COMPARISON OF THE EFFECTIVITY

OF WIND SPEED FORECASTING METHODS

Diksha Kaur

A thesis submitted to

Auckland University of Technology

in fulfillment of the requirements for the degree of

Doctor of Philosophy (PhD)

2015

School of Engineering

Primary Supervisor: Tek Tjing Lie

II

Abstract

Environmental considerations and reducing carbon emission has accelerated the use of

various renewable resources for electricity generation. Wind generation, in this context has

seen sustained increases globally. Wind intermittency, its independent nature of direction,

and varying speed are the best-known challenges and major barrier for accommodating very

larger wind power penetration. There are several factors that can address this and help

improve the attractiveness of the wind power to a utility. These may include improvements

in model accuracy to decrease the forecast error, changes in the conventional plant, better

storage or better load management to name a few.

Managing the wind energy intermittency for existing power system operation and control

therefore becomes crucial. The issues posed in the wind speed prediction include reduction

in time delay, improvement in speed prediction for short time, error reduction, model

improvement for effective conversion of wind energy. However there is a lot of research

being done in this field in which individual as well as hybrid forecasting techniques are

being worked upon.

One effective solution is to predict the future values of wind power production, which is

usually dependent on the wind speed. Precise forecasting of wind speed is vital to the

effective harvesting of wind power. So, if an accurate forecasting of the wind speed from a

few minutes to several hours ahead is obtained, the effective integration of larger penetration

of the wind power generation can be achieved efficiently, safely and economically.

The main objective of this research is to compare the effectiveness of different techniques

and to come out with a few unique hybrids that help reduce the forecasting error. The main

focus lies on error reduction and improvement of model by hybridizing different techniques.

It focuses on the improvement of the present forecasting methods and reduction of the

forecasting error. This thesis presents a critical literature review and an up-to-date

bibliography on the wind forecasting technologies. One of the objectives of this research is

to develop a few novel wind speed forecasting techniques, which produce more accurate

prediction.

III

Initially a hybrid technique, Artificial Neural Networks (ANN) along with the statistical

method Ensemble Kalman Filter (EnKF) is proposed. The proposed method is used for a

short-term prediction of wind speed. The well-established MATLAB software computing

environment is utilized to simulate and show the effectiveness of the proposed hybrid

technique. By help of past observations of wind speed, the EnKF is found to correct the

output of ANN to find the best estimate of wind speed. The simulation results in MATLAB

show that combination of ANN with EnKF acts as an output correction scheme.

For the next hybrid technique, the Wavelet Transform (WT) along with the Auto Regressive

Moving Average (ARMA) is proposed. In the preliminary stage of the investigation, this

combination is expected to give minimum Mean Absolute Percentage Error (MAPE). A

simple simulation study has been conducted by comparing the forecasting results using the

Wavelet-ARMA with the ANN-EnKF hybrid technique to verify the effectiveness of this

new proposed hybrid method. The simulation results of the proposed WT-ARMA hybrid

technique show significant improvements in the forecasting error.

The thesis has also investigated how to fully utilize Auto Regressive Moving Average

(ARMA) to predict wind speed. However, the order estimation of ARMA is a very critical

issue. Therefore, ANN has been used for parameter estimation, which is then combined with

the Akaike Information Criteria (AIC) for order estimation. A simulation study has been

conducted by comparing the proposed hybrid results with the Genetic Algorithm (GA) for

parameter estimation and an exhaustive search for order estimation.

Another part of this research focuses on the Economic Dispatch (ED) problem. The

stochastic nature of the wind and the highly nonlinear transformation from wind speed to

electrical energy makes it more difficult to determine how to dispatch its power in order

to guarantee both operational cost reduction and power system security. From a network

constraint perspective in the economic dispatch problem, one of the factors to be accounted

for is voltage instability, which impacts both active and/or reactive power dispatch. As a

solution, an Optimal Reactive Power Dispatch (ORPD) based on Particle Swarm

Optimization (PSO) using Graph Theory (GT) has been proposed to overcome the

aforementioned problem. The Graph Theory has been proposed since it is very useful in

IV

cases of fault detection and isolation or to shed unbalanced nodes in case of excessive or

insufficient supply. Simulation studies on the IEEE-14 Bus System have been conducted to

show the effectiveness of the proposed method.

The research utilizes real time data from a few minutes to several days ahead to show the

effectiveness of the proposed methods. Both synthetic data and real information from New

Zealand wind resources have been used. Since the forecasting errors may vary within the

time frame under consideration, different modifications have been added to the proposed

hybrid techniques to get robust results from the practical data. In addition, the correlation

between the forecasting error indices and the economic factors has been investigated. The

Economic Dispatch problem has been identified as a major one and forecasting has been

used as a solution in Graph Theory. This research may not only benefit forecasting of wind

but also several other applications as well such as load forecasting or price forecasting in the

future.

V

Table of Contents

Abstract........................................................................................................................ II

Table of Contents............................................................................................... V

List of Figures....................................................................................................IX

List of Tables......................................................................................................X

List of Abbreviations Used................................................................................XI

Attestation of Authorship.................................................................................XV

Acknowledgements ........................................................................................XVI

List of Research Outputs .............................................................................. XVIII 1.Introduction...............................................................................................1

1.1 Background and Rationale of the study ............................................................. 1

1.2. Problem Statement & Research Questions ..........................................................4

1.2.1Research Questions.......................................................................................... 4

1.2.2 Problem Domain.............................................................................................. 4

1.3. Aims and Objectives of the Study ..........................................................................6

1.4. Organization of the Thesis..................................................................................... 10

2. Literature Review ................................................................................................12

2.1 Introduction ................................................................................................................12

2.2 Different types of forecasting errors ....................................................................12

2.3Wind Speed Forecasting Techniques…………………………………...….14

2.3.1 Very-Short Term Forecasting ..................................................................14

2.3.2 Short Term Forecasting..............................................................................15

VI

2.3.3 Medium Term Forecasting........................................................................16

2.3.4 Long Term Forecasting .............................................................................17

2.3.5 Data Preprocessing .....................................................................................19

2.3.6 Design and training of forecast models .................................................19

2.3.7 Selection of number of input nodes and output....................................20

2.3.8 Implementation and Validation of forecast models ............................20

2.3.9 Persistence Method ....................................................................................20

2.3.10 Physical Approach.....................................................................................21

2.3.11 Statistical Approach (ANN and Time series models) ......................22

2.3.12Hybrid Approach .......................................................................................23

2.3.12.1 ANN-EnKF hybrid.................... .............................................24

2.3.12.2 ARMA-WT hybrid.................................................................. 26

2.3.12.3 ARMA-ANN-AIC hybrid............. .......................................28

2.3.12.4 ORPD-PSO-GT solution for the Economic

Dispatch problem............................................. ............................29

2.10 Summary ...................................................................................................32

3. ANN-EnKF Hybrid......................................................................................... 34

3.1 Introduction ................................................................................................................34

3.2 Surrogate Of ANN ...................................................................................................34

3.3 Ensemble Kalman Filter......................................................................................... 36

3.4 Hybridizing ANN Surrogate and Kalman Filter ...............................................37

3.5 Results and Discussion........................................................................................... 38

3.6 Summary.................................................................................................................... 45

4. WT-ARMA Hybrid..............................................................................................46

4.1 Introduction................................................................................................................ 46

VII

4.2 Wavelet Analysis ......................................................................................................47

4.2.1 Continuous Wavelet Transform (CWT) ..............................................49

4.2.2 Discrete Wavelet Transform (DWT) and Multi Resolution

Analysis MRA)................................................................................................... 50

4.3 ARMA..........................................................................................................................52

4.4 Results and Discussion.............................................................................................54

4.5 Summary..................................................................................................................... 59

5. ARMA-ANN-AIC Hybrid...................................................................................60

5.1 Introduction ................................................................................................................60

5.2 ARMA .........................................................................................................................60

5.3 ANN ............................................................................................................................61

5.4 AIC ..............................................................................................................................62

5.5 Results and Discussion .............................................................................................................64

5.6 Summary........................................................................................................................................... 68

6. ORPD and PSO in GT ......................................................................................... 69

6.1 Introduction................................................................................................................ 69

6.2 Optimal Reactive Power Dispatch ........................................................................70

6.3 Particle Swarm Optimization (PSO) .......................................................... 72

6.4 Wavelet Transform and ARMA Hybrid..............................................................74

6.5 Graph Theory (GT) ..................................................................................................75

6.6 Results and Discussion …....................................................................................77

6.7 Summary.....................................................................................................................81

7. Conclusion and Future Research Recommendations................................ 82

7.1 Conclusion................................................................................................................. 82

VIII

7.2 Future Research Recommendations ..................................................................................84

References .................................................................................................................85

IX

List of Figures

Figure 3.1 Schematic presentation of ANN surrogate

Figure 3.2 Original Wind Data

Figure 3.3 Output of ANN fed with original wind data

Figure 3.4 Output of Ensemble Kalman Filter

Figure 3.5 Output for the hybrid of ANN and EnKF.

Figure 3.6 Error Graph for ANN-EnKF Hybrid

Figure 3.7 Root Mean Square Error for various wind speeds

Figure 4.1 Procedure of Wind Speed Forecasting using WT-ARMA

Figure 4.2 Output of Wavelet Analysis shows approximations for the low

frequency (A5) and details for high frequency Dj (1≤ j≤ 5) that are

added as (S).

Figure 4.3 Output of ARMA fed with Wavelet

Figure 4.4 Actual Wind Speed versus Forecasted using Hybrid Model

Figure 4.5 Error graph (MAPE) for WT-ARMA hybrid

Figure 5.1 A three layer ANN topology. Note that the weights of x input neuron

are W leads and the weights of y input neurons are V leads.

Figure 5.2 Procedure of the ANN-ARMA-AIC hybrid

Figure 5.3 Difference between the parameters for ANN and GA

Figure 6.1 Flowchart showing the sequence of problems in ED

Figure 6.2 Standard IEEE- 14 bus system with DFIG at Bus 3

Figure 6.3 Graph showing the relationship of Power Flow

Figure 6.4 Total Reactive Power Losses

X

LIST OF TABLES

Table 2.1 Time Scale and Error Classification for Different Technique

Table 2.2 Basic Wind Forecasting Methods

Table 3.1 Root mean square errors for different wave ensembles

Table 4.1 Different Error for Different values of 1 ≤ j ≤ 6

Table 4.2 Comparison between Wavelet ARMA hybrid and ANN-KF in Wind

Speed Forecasting

Table 5.1 The Exhaustive Search Results For Different p And q

Table 5.2 Comparison Of Parameters Using ANN And GA

Table 5.3 AIC Values

Table 6.1 Total Reactive Losses

XI

List of Abbreviations Used

AIC Akaike Information Criteria

ARIMA Auto Regressive Integrated Moving Average

ARCH Auto Regressive Conditional Heteroscedasticity

AR Auto Regressive

ARMA Auto Regressive Moving Average

ANFIS Adaptive-Network-based Fuzzy Inference Systems

ANN Artificial Neural Networks

AWNN Adaptive Wavelet Neural Network

BJ Box–Jenkins

CWT Continuous Wavelet Transform

DE Differential Evolution

DFIG Double Fed Induction Generator

DWT Discrete Wavelet Transform

ED Economic Dispatch

EMD Empirical Mode Decomposition

EMS Energy Management System

EnKF Ensemble Kalman Filter

ENVISAT Environmental Satellite

EP Evolutionary Programming

XII

ESACF Extended Sample Auto Correlation Function

FFNN Feed Forward Neural Network

GA Genetic Algorithm

GARCH Generalized Auto Regressive Conditional Heteroscedasticity

GT Graph Theory

GWN Gaussian White Noise

HDE Hybrid Differential Evolution

HIRLAM High Resolution Limited Area Model

HMPSO Hybrid Multi-Swarm Particle Swarm Optimization

HONN High Order Neural Network

KF Kalman Filter

LMBP Levenberg–Marquardt Back Propagation

MA Moving Average

MAE Mean Absolute Error

MAPSO Multi Agent Particle Swarm Optimization

MAPE Mean Absolute Percentage Error

MATLAB Matrix Laboratory

MCE Maximum Co Entropy

MEE Minimum Error Entropy

MEEF MEE with fiducially points

MLP Multi-Layer Perceptron

XIII

MPE Mean Percentage Error

MERIS Medium Resolution Imaging Spectrometer

MPSO Modified Particle Swarm Optimization

MRA Multi Resolution Analysis

MSPE Mean Squared Prediction Error

NIWA National Institute of Water and Atmospheric Research

NLP Non-Linear Programming

NWP Numeric Weather Prediction

OPF Optimal Power Flow

ORPD Optimal Reactive Power Dispatch

P Persistence Method

PFNN Polynomial Function Neural Network

PSO Particle Swarm Optimization

RBf Radial Basis Function

RMSD Root-Mean-Square Deviation

RNN Recurrent Neural Network

SAR Synthetic Aperture Radar

SCIG Squirrel Cage Induction Generator

SFLA Shuffled Frog Leaping Algorithm

SOA Seeker Optimization Algorithm

SSM/I Special Sensor Microwave/Imager

XIV

STFT Short-Time Fourier Transform

SVM Support Vector Machine

WEP Weather Ensemble Predictions

WT Wavelet Transform

WTG Wind Turbine Generators

XV

Attestation of Authorship

I hereby declare that this submission is my own work and that, to the

best of my knowledge and belief, it contains no material previously

published or written by another person (except where explicitly

defined in the acknowledgements), nor material which to a

substantial extent has been submitted for the award of any other

degree or diploma of a university or other institution of higher

learning.

Diksha Kaur

XVI

Acknowledgements

First and Foremost, I would like to thank the Almighty, “WAHEGURU” who has always

been “Gursahai”, which means that he has always been in me, around me and with me. I am

truly blessed to be a “Sikh” who always has the passion to learn whatever comes their way.

I am very grateful to my primary supervisor Tek Tjing Lie from Auckland University of

Technology, who gave me this opportunity to work under him. He has always helped me

extensively to develop academic research skills during my study period. I really appreciate

his valuable guidance and sympathetic encouragement, due to which the project has been

fulfilled smoothly.

I wish to express gratitude to Nirmal C Nair, my secondary supervisor from the University

of Auckland, who has generously made helpful comments, creative suggestions and

contributive discussions from the first to the final level of my thesis.

I am very thankful to my co-supervisor James L Kirtley Jr. from Massachusetts Institute of

Technology, who gave me the opportunity of working under him and completing the second

half of my research in Boston. I really am grateful to him for investing his precious time in

my study.

I must admit I cannot imagine how this thesis would look like if I did not have Brice Valles

as a co-supervisor. I feel I have learnt a lot from him and I am very grateful for all the time

he spent on guiding me, for his patience with me and for the continuous support he gave me.

His personal kindness and scientific efforts has inspired me throughout the course of my

research.

I would like to dedicate this thesis to my dear parents, Rajesh Singh and Archna Kaur who

have always been there to support me through thick and thin. My father has been my source

of inspiration and has helped and worked with me throughout this journey. A very special

thanks to my sister, Reeti Kaur and brother in law, Manpreet Singh who were with me from

the start encouraging me to take up on this opportunity. Without their blessing, this thesis

XVII

would have not been possible. My heart goes out for all the kids in the family Gursahai,

Neev, Arshliv, Haramrit and Japjyot whose smiles have always made things look very

simple.

I would like to acknowledge and especially thank my loving, caring and supportive husband,

Sunmeet Singh. His patience, love and enthusiasm over the duration, has been my pillar of

strength throughout.

Many thanks are given to the data provider NIWA. NIWA is a crown-owned research and

consultancy company, with a global reputation as experts in water and atmospheric research,

http://www.niwa.co.nz

Last but not the least, I would like to thank the Auckland University of Technology for

providing a Doctoral Scholarship for the work achieved in this research.

XVIII

List of Research Outputs

1. D. Sharma, T. T. Lie (2012) Wind Speed Forecasting Using Hybrid

ANN-Kalman Filter Techniques. Paper presented at the international

symposium on the International Power and Energy, 14-16 Dec 2012.

2. D. Sharma, T.T. Lie, N. K. C. Nair, B. Valles (2015) Wind Speed

Forecasting using ANN, ARMA and AIC hybrid to Ensure Power Grid

Reliability. Paper presented at the international symposium on the

Innovative Smart Grid Technologies (ISGT) Asia, 4-6 Nov 2015.

3. D. Kaur, T.T. Lie, N. K. C. Nair, B. Valles, “Wind Speed Forecasting

Using Hybrid Wavelet Transform-ARMA Techniques,” AIMS Energy,

vol. 3, pp. 13-24, 2015.

1

Chapter 1

Introduction

1.1 Background and Rationale of the study

Environmental considerations have prompted the use of various renewable sources for

electricity generation. It has led to a strong interest and an increase in the use of various

renewable resources. Wind is a clean, renewable resource that may be converted into

electrical power and has become very popular in recent years. With the increasing popularity

and utilization of wind power technology, the cost of the new energy resources has already

approached the conventional energy and thus, it has the potential ability to compete with

traditional power plants [1].

Wind energy is one of the fastest growing energies and it is becoming more economically

attractive causing reduction in the operating cost of any utilities. Wind speed and power

predictions are beneficial for stabilization of power, matching demand and supply and power

grid management [2].

But the main problem with wind is its intermittent nature that makes the output power of

wind power generations difficult to control [1]. It becomes the greatest challenge while

integrating wind power into the electric grid [8]. This nature of the wind results towards

increasing the complexity of the existing power system regulation and reserves requirements

(also known as ancillary services in electricity markets) that are used to maintain stability

and reliability. Managing the intermittency of wind towards existing power system operation

2

and control therefore becomes crucial. The intermittency of wind has a greater impact on

grid security, system operation, and market economics [3].

Reference [2] shows that at wind penetrations of up to 20% of system peak load, system-

operating cost, such as unit commitment cost, would increase arising from wind variability

and uncertainty. The world installed capacity of wind power generation has increased from

60 GW in year 2000 to 160 GW in June 2010, and it is estimated to be 460 GW by the end

of year 2015 [3]. The present total consumption of electricity from wind energy in New

Zealand is close to 5% and is estimated to go up to 20% by 2030 as reported by some

electricity generation and retailer companies including Genesis Energy, Meridian and

Mighty River Power according to the Business and Economic Research Limited (BERL)

report [4]. Because wind variability and uncertainty make the output power of wind farms

difficult to control, an increase of wind penetration will result toward increasing needs for

power systems regulation and reserves requirement to ensure stability and reliability.

One effective solution is to predict the future values of wind power production. The most

important factor responsible for wind power generation is the local wind speed. Since wind

power is a function of wind speed, forecasts of power are generally derived from forecasts of

wind speed. So, if a reliable forecasting is done of wind speed/power from a few minutes to

several hours ahead, the effective utilization of wind power and generation of electricity can

be obtained thereby relieving the electricity network from various shortcomings and stress

associated with uncertainty [6].

The increase on wind power generation requires solutions to a number of research fields

including competitive market designs, real-time grid operations, interconnection standards,

quality of power, ancillary service requirements and costs, capacity of transmission system

and its future upgrades, stability and reliability of power system, optimal reductions in

greenhouse gas emissions of entire power system (determined by optimal amount of wind

penetration into system). Forecasting has become a very vital part for planning in areas

where there is high concentration of wind generation and a limited transmission capacity of

network. For instance, forecasting may help understand all the imbalances of the electricity

market well in advance. Also, it will enable develop well-functioning hour-ahead or day-

ahead markets. Another example is that even though wind energy is not dispatched, like

other conventional generation, the cost of electricity may be reduced drastically if wind

3

energy can be scheduled using accurate wind forecasting [11].

With increasing wind penetration more robust and innovative approaches are required to

decrease the forecasting error and improve the generation output estimation model. Several

prediction methods have been discussed in the past few years and several studies have been

performed to develop the precision of wind speed prediction.

There is clearly a requirement for accurate wind forecasting in order that wind power can be

integrated into the scheduling and dispatch decisions of the power system operators. Due to

turbulent wind environment, wind speed and wind power vary and output power of wind

turbine generator fluctuates due to the same. It can be marginally improved with changes

that could improve the model and reduce some errors in the prediction.

There are several factors that can be taken care of to improve the attractiveness of wind

power to a utility. These include improvements in model accuracy to decrease the error in

wind forecasting, changes in the conventional plant mix such as using fast response plants,

shorter start-up and ramping times for thermal plants, better storage, better load management

to accommodate fluctuations [1].

As a solution, more sophisticated and innovative approaches are required. Therefore,

different hybrid approaches are formed along with the traditional statistical techniques to

give effective and required results. Combination of different methodologies such as mixing

physical and statistical approaches or combining short-term and medium-term models,

combination of alternative statistical models, to name a few, is referred to as a hybrid

approach. Many types of hybrid models are utilized to predict wind power [5-6]. The aim of

hybrid models is to benefit from the advantages of each model and obtain a globally optimal

forecasting performance.

Among many hybrid techniques, the statistical time series and neural network methods are

mostly aimed at short-term predictions. In several predictions, they use the difference

between the predicted and actual wind speeds in the immediate past to tune the model

parameters. Of volatility and deviation average reported error for the wind forecasting is in

the order of 10%~20% of the installed power for a 24 hour horizon [11].

4

1.2. Problem Statement & Research Questions

1.2.1 Research Questions

To strengthen the concept, promising research questions have been identified as

follows:

Which hybrid methodology is suitable for wind speed forecasting to get the

best accuracy?

How can the proposed hybrid be extended to predict from 1 hour ahead to 6

hours ahead?

What is the impact on the economics?

To achieve the success of this study, the following objectives will be targeted:

Investigation and analysis of the existing methodologies and developing a

new hybrid model that compensates for the forecasting error.

Development of a method that can be extended from a few minutes to several

days of forecasting and is still able to deliver an output that has minimum

error.

Investigate the correlation between forecasting error indices and the economic

impacts.

1.2.2 Problem Domain

Forecasting has been performed for power systems such as load forecasting or weather

forecasting. There exist various popular methods including Artificial Neural Network

(ANN), Kalman Filter (KF), Auto Regressive Moving Average (ARMA), Auto Regressive

Integrated Moving Average (ARIMA), Auto Regressive Conditional Heteroscedasticity

(ARCH), Generalized Auto Regressive Conditional Heteroscedasticity (GARCH), Adaptive-

Network-based Fuzzy Inference Systems (ANFIS), Grey Predictors just to name a few.

5

However, these methods are not able to decrease the forecasting error further individually so

research on their hybrids variations are becoming more popular.

The advantages of these methods are different depending on what methodology is utilised.

Few of these methods are very common, easy to use and easy to understand. Few methods

require less statistical training whereas others require more statistical training. Most of these

methods are widely used for forecasting purposes. There are others that can either fit the

approximation to any time series or are able to handle missing data very efficiently. In

addition, there are some methods prone to over fitting. Few of them can include random

noise, which can make it very difficult to use whereas some may assume the system as a

linear system.

To increase the capability of these individual methods, hybridization is usually done. It has

been tried by various researchers and has also resulted in some advantages and

shortcomings. The shortcomings are as follows: (i) some of the hybrids have worked well in

terms of time consumption, (ii) some hybrids can be very time consuming, (iii) some hybrids

have not been able to decrease the forecasting error as per the demand, (iv) few hybrids have

been able to predict very short term forecasting fairly and (v) some hybrids have been tried

for medium term forecasting but have not been proved as successful as required. The dire

need is to take forecasting to another level and predict at least up to six hours accurately.

Also, some of the methods that have been recently introduced either give large error or incur

high computational time.

These advantages and pitfalls have led this research to propose a solution that is not only

time efficient but also is able to decrease the forecasting error. Since the impact of better

accuracy on power systems is very significant, there is a lot of interest in developing

techniques that are novel.

The main problems identified include: (i) the intermittent nature of wind has to be captured

by some forecasting techniques so that effective utilization of wind energy is obtained; (ii) a

method is required that fits as much of the environment data and surrounding changes as

possible and be flexible. The chosen method should also enable accurate estimation and be

quickly computed.

6

1.3. Aims and Objectives of the Study

This thesis provides an insight of all the categories of wind speed prediction methods,

associated with wind power and speed, their classification with respect to time horizons,

different ranges, and different approaches based on numeric weather prediction (NWP),

statistical approaches, artificial neural network (ANN) and hybrid techniques over different

time-scales. It begins by presenting a critical literature review and an up-to-date bibliography

on wind forecasting technologies. Various issues have been highlighted primarily related to

wind speed and thereby power generation. From all the available methods used in prediction

mentioned in the literature review, this thesis mainly focuses on the improvement of the

forecasting methods and reduction of the forecasting error when applied to practical and

realistic data. The choice of the wind model used amongst the various existing models is

based on several performance parameters. Reference [7] gave a detailed survey of the wind

speed forecasting techniques that have been provided since the past 15 years.

The first objective of this research is to critically assess two methods for wind speed

forecasting namely Artificial Neural Networks (ANN) and Ensemble Kalman Filter (EnKF)

followed by formulating a hybrid method and assess its performance practically using

MATLAB and bring down the forecasting error to a minimum. The short-term prediction

methods using hourly prediction have been done in this thesis. An ANN is normally used to

map random input vector into the corresponding output vector. A surrogate is constructed

based on modeling the response of large-scale model to a limited number of intelligently

chosen data points. EnKF is applied on wind-wave data. It will then correct the output of the

ANN to find the best estimate of the wave height. Coupling this surrogate (instead of a full

model) with EnKF leads to computational efficiency. All these are tested under MATLAB

developed programs to compare the individual results with the hybrid to see what effect the

hybrid has. References [8-10] highlight EnKF method resulting in relatively minor errors of

approximately 3.97% and therefore higher prediction precision has been obtained in this

hybrid.

For the second objective, the preliminary investigation on the Wavelet Transform (WT)

along with the ARMA model is proposed to form a hybrid method for better forecasting. The

Wavelet analysis is a well-established technique that is being adapted in various fields for

forecasting. It analyses the signal in nature and the time frequency. The WT decomposes and

7

then reconstructs the historical wind speed data into various factors. The second method

used is the ARMA model, which is a time series model. The MAPE of the ARMA has been

shown to be between 7% for 1-hour up to 18% for the next six hours [11]. The aim was to

decrease the MAPE to around 5% in this research.

Reference [12] shows that a combination of methods has been successful for predicting

water demand of Dalian City. It combines WT, ARMA and Least Squares methods and

comes out with the most accurate prediction of that area. This hybrid however, has not been

tried much in the field of wind forecasting yet. Reference [13] shows the potential of this

hybrid but this paper fails to justify it for a long duration of time and is very limited on

experimental observations. Reference [14] shows the capability of the hybrid method by

evaluating few case studies and proposing recommendation for further research. It

demonstrates how pre -filtering can enhance the performance and efficiency of computation.

This thesis has investigated how a hybrid method using WT and ARMA can be used towards

wind speed forecasting.

The Wavelet analysis is a well-established technique used in various application areas for

forecasting. It analyses the signal capturing both time and frequency resolutions. It also

overcomes the drawbacks of Short-Time Fourier Transform (STFT) as it is a windowing

technique that uses variable sized regions [14]. The second techniques used for the proposed

hybrid is the ARMA model, which is a time series model. ARMA models are widely used

because of their simplicity, cost-effectiveness and accuracy for timely forecasting. It is used

to solve problems in the fields of mathematics, finance and engineering industry that deal

with a large amount of observed data from history.

Another part of this research is to develop a novel wind speed forecasting technique, which

produces more accurate predictions. A hybrid method composed of the Artificial Neural

Network (ANN) and Akaike Information Criteria (AIC) along with the Auto Regressive

Moving Average (ARMA) technique is proposed. Simulation studies were conducted to

show the effectiveness of the proposed method. The simulation study results of the proposed

hybrid have shown very good results through reduced forecasting error for the test data used

in this thesis.

8

One of the main challenges is to properly determine the order of ARMA that would optimize

the results. To find the optimum ARMA order, another method has to be used. Instead of

the most basic Box–Jenkins (BJ) identification (which requires all calculations to be

repeated when new pieces of data arrive [15]) or similar methods, this research comes out

with a very novel identification method that uses ANN and AIC to determine the order of

ARMA. The comparisons are made with the Genetic Algorithm (GA) method and are based

on the value of the coefficients obtained. The results show that the ANN computes the

coefficients of an ARMA system accurately. This part uniquely combines ANN with AIC to

determine the true order of ARMA and bring down the forecasting error. Consequently, the

proposed method will not only benefit the wind forecasting community but also can be

adapted in the future towards load forecasting and price forecasting methods in power

systems.

The third and last objective is to find the correlation between the forecasting and its

application in power systems. Therefore, another problem identified with wind power i.e.,

the Economic Dispatch (ED) has been explored. In the modern Energy Management System

(EMS), ED serves a vital functionality. Wind power may be dispatched above or below its

available capacity during periods of excessive or insufficient supply. During excess

generation periods, to avoid experiencing additional shutdowns and start-ups of conventional

plants economic curtailment of renewable energy could be the most cost-efficient dispatch

option. When there is a surplus, the extra generated wind energy should be stored otherwise

would not be fully utilized. Many companies are working on utility-scale storage units that

they hope will help balance intermittent renewable energy

The objective of ED is to schedule the power generation properly in order to minimize the

total operational cost. It is usually quite difficult to determine how to dispatch the power in

order to guarantee both operational cost reduction and power system security. Therefore, it is

highly desirable to investigate how to dispatch the power in an appropriate and

economic manner for the power system. So, in order to make accurate dispatch decisions it

becomes crucial that the wind forecast is accurate [16].

Exact forecasting of the expected generation from a wind farm is nearly unmanageable

predominantly due to the stochastic nature of the wind along with the highly nonlinear

9

transform from wind speed to electrical energy. Due to forecasting errors, wind is not

dispatched appropriately or is dispatched below the available capacity. Thus, the main

problem here lies in accommodating wind energy scheduling to existing Economic Dispatch

(ED) application. One of the constraints of the economic dispatch formulation is voltage

stability, which depends primarily on reactive power dispatch availability. The reactive

power is generally automatically available due to synchronous machine field control and

transmission line characteristics. However, it affects the total generating cost by increasing

the transmission losses [17]. Reactive Power Optimization is a mixed integer nonlinear

programming problem where meta-heuristics techniques have proven suitable for providing

optimal solutions [18]. Therefore, in order to stabilize voltage and dispatch it properly,

Optimal Reactive Power Dispatch (ORPD) is required to control the regulating equipment to

optimize reactive power flow, reduce active power and voltage losses, improve voltage

quality and to make electric equipment work safely and reliably.

ORPD is a key instrument to achieve secure and economic operation of modern power

systems [19]. Due to the complex characteristics of ORPD, heuristic optimization has

become an effective solver. In this research, the Particle Swarm Optimization (PSO)

approach is used to solve the ORPD problem for ED and has been tested on the IEEE-14 Bus

System using Graph Theory (GT) to get the results for reactive power optimization. The

objective of the proposed PSO is to minimize the total support cost from generators and

reactive compensators. It is achieved by maintaining the whole system power loss as

minimum thereby reducing cost allocation. The GT shows the relationship between the

forecasted wind speed and its impact on the economics.

In the past decade, heuristic optimization algorithms, such as genetic algorithm (GA) [20-

21], PSO [22-23], just to name a few have been applied in reactive power optimization. PSO

has some advantages over other similar optimization techniques such as GA. PSO has a

more effective memory capability than the GA since every particle remembers its own

previous best value as well as the best neighborhood.

Moreover, PSO is easier to implement and there are fewer parameters to adjust. PSO is more

efficient in maintaining the diversity of the swarm (more similar to the ideal social

interaction in a community), since all the particles use the information related to the most

10

successful particle in order to improve themselves, whereas in GA, the worse solutions are

discarded and only the good ones are saved; therefore, in GA the population revolves around

a subset of the best individuals [24].

GT is an important tool, which can be used in a lot of research areas of computer science

such as data mining, image segmentation, clustering, image capturing, networking and others

[25]. Problems of efficiently planning routes for finding the shortest path in a network,

searching an element in a data, fault detection and isolation, shed unbalanced nodes may be

managed very easily in GT and hence it becomes a very beneficial tool.

This research is tested on a standard IEEE-14 bus system on GT where double fed induction

generator (DFIG) wind turbines are used. DFIG are used typically for high power wind

generation systems (>1.5 MW) in several countries. DFIG have the distinct advantage of

being able to generate controllable power by reduced rated power converters in comparison

with other wind generator technologies for the same power [26-27]. DFIG based system is

more suitable to integrate into power system for wind power generation than Squirrel Cage

Induction Generator (SCIG) system from viewpoint of power system small signal stability.

1.4. Organization of the Thesis

This thesis consists of seven chapters. Chapter 1 provides an overview of the background of

the study, followed by the problem statement, research questions, significance of the study,

and finally organization of the thesis.

Chapter 2 provides an extensive review of literature related to the concepts of this study,

namely: types of errors, traditional techniques, and state of the art methods. It describes the

literature analysis giving the main areas of intersection of ANN-EnKF, ARMA-WT and

ARMA-ANN-AIC hybrids respectively.

Chapter 3 covers the research approach for the ANN-EnKF hybrid. This includes research

philosophy, research method of the hybrid technique and its effectiveness. It also discusses

the results in detail.

11

Chapter 4 explains the ARMA-WT hybrid in detail. It comes out with a unique hybrid

equation that combines the main mathematical equation of both the individual methods. It

also explains the results that have been tested on MATLAB.

Chapter 5 focuses on the ARMA-ANN-AIC hybrid which is used to determine the

parameters estimation to find the true order for ARMA. It also gives the methodologies and

results.

Chapter 6 provides a detailed discussion of the economic dispatch problem. It starts with an

overview of the inductive approach and follows with a discussion of ORPD implemented on

PSO in GT.

Chapter 7 summarizes the study with a recapitulation of what has been achieved, an overall

discussion of the findings, discussions of the study’s implications and of its limitations, and

provides suggestions for future study.

12

Chapter 2

Literature Review

2.1 Introduction

This Chapter presents a critical literature review analysis on wind forecasting methods and

related topics. The review covers a wide range of recent literature in the problem domain and

is classified based on forecasting errors, hybrid methods, forecasting approaches just to

name a few.

2.2 Different types of forecasting errors

A forecast error is the difference between the actual or real and the predicted or forecast

value of a time series or any other phenomenon of interest. There are various types of errors

such as the mean percentage error (MPE), root-mean-square deviation (RMSD), mean

squared prediction error (MSPE), mean absolute error (MAE), Mean Absolute Percentage

Error (MAPE), and so on. Reference [28] explains the errors, their relevance, different types

of forecasting techniques and the latest developments in the ANEMOS report.

The mean percentage error (MPE) is the computed average of percentage errors by which

forecasts of a model differ from actual values of the quantity being forecasted. A

disadvantage of this measure is that it is undefined whenever a single actual value is zero.

The MPE gives an indication of how good a measurement is relatively to the size of the

thing being measured [29]

𝑀𝑃𝐸 =100%

𝑛∑

𝑓𝑡−𝑎𝑡

𝑎𝑡

𝑛𝑡=1 (2.1)

13

where at is the actual value of the quantity being forecasted, ft is the forecast, and n is the

number of different times for which the variable is forecasted.

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently

used measure of the differences between values predicted by a model or an estimator and the

values actually observed. These individual differences are called residuals when the

calculations are performed over the data sample that are used for estimation, and are called

prediction errors when computed out-of-sample. The RMSD serves to aggregate the

magnitudes of the errors in predictions for various times into a single measure of predictive

power. RMSD is a good measure of accuracy, but only to compare forecasting errors of

different models for a particular variable and not between variables, as it is scale-dependent

[3]. For an unbiased estimator, the RMSD is the square root of the variance, known as the

standard error. The RMSD of predicted values ��𝑡 for times t of a regression's dependent

variable y is computed for n different predictions as the square root of the mean of the

squares of the deviations:

𝑅𝑀𝑆𝐷 = √∑ (𝑦𝑡 − ��𝑡)

2𝑛𝑡=1

𝑛 (2.2)

The mean squared prediction error (MSPE) of a smoothing or curve fitting procedure is the

expected value of the squared difference between the fitted values �� and the (unobservable)

function g. If the smoothing procedure has operator matrix L, then

𝑀𝑆𝑃𝐸 (𝐿) = 𝐸 [(𝑔(𝑥𝑖) − �� (𝑥𝑖))2 ] (2.3)

The mean absolute error (MAE) is a quantity used to measure how close forecasts or

predictions are to the eventual outcomes [30-31]. It is the average of the absolute errors,

where fi is the prediction and yi the true value.

𝑀𝐴𝐸 =1

𝑛∑ |𝑓𝑖 − 𝑦𝑖|

𝑛𝑖=1 (2.4)

14

The MAPE is a measure of accuracy in fitted time series value. The absolute values of all

errors are summed and the average is computed. It expresses accuracy as a percentage. It has

the same units as the original data. It can take a range of values between zero to infinity [1,

32]. This is the reason why it has been largely used in wind speed forecasting by various

researchers. MAPE stems from the law governing that wind farm owners who want to

participate in the electricity market have to predict their own power. Deviations from the

declared schedule are punished according to this error measurement [28]. This thesis also

uses MAPE to forecast wind speed due to its accuracy.

𝑀𝐴𝑃𝐸 =1

𝑛∑ |

𝐴𝑡−𝐹𝑡

𝐴𝑡|𝑛

𝑡=1 (2.5)

where At is the actual value and Ft is the forecast value.

Review of wind speed forecasting techniques has been done on the basis of time scale,

design issues and types of forecasting techniques in the following section.

2.3 Wind Speed Forecasting Techniques

This section describes different forecasting techniques based on time-scales, design issues,

individual and hybrid methods.

2.3.1 Very-Short Term Forecasting

Very short term is defined as the prediction from a few seconds to half an hour. This is

because the deregulated markets are cleared almost every 5 min and settlements are done

every 30 min. This is shown in [33] and [34] for 5 to 15 minutes timeframe. Tests are

conducted and wind power data are smoothed for 30 sec resolution to match the resolution of

the wind speed. Results show that wind power variation somewhat matches that of wind

speed.

The Ensemble Kalman Filter (EnKF) method is explained in [35]. Reference [36] uses it to

predict 10 min ahead wind speed. The ARMA model is used as the state function of the

EnKF and perturbed initial wind data are taken into consideration to generate the ensemble.

15

The comparison between the methods shows that the EnKF is suitable for wind speed

prediction and improve grid integration of wind energy and gives a MPE of about 5.06% but

only for 10 min ahead forecasts.

The most common method used is the Prediction Method whereas others include ANN,

ARMA, Auto Regressive Integrated Moving Average (ARIMA), Adaptive-Network-based

Fuzzy Inference Systems (ANFIS) and so on. These are all used individually. A few papers

are available for very-short term forecasting timeframe. Linear prediction model of the

ARMA is shown in [34] where the output is a combination of past and present values. At

low frequency wind speed is filtered from a low pass filter and results from 1s to 5s are

reported. The time frame taken in this paper is not very big and few results have been shown.

2.3.2 Short Term Forecasting

Short term forecasting refers to the prediction from half an hour to six hours. Most research

on wind forecasting has been done for this time scale. If we are able to predict the next 6

hours accurately it becomes very helpful in proper utilization of wind energy. There are

numerous combinations tried in this timescale but even then there is still a lot of scope left

since the error is still relatively significant.

The term wind pattern as proposed by [33] and [34] characterizes different types of trends of

wind and different time duration. The trends are combined with the ARMA with only

historical generation data for 1-hour advance parameters. A wind speed pattern is identified

and its model is established. Wind speed data of a wind farm in Inner Mongolia, verifies it

giving a small MAPE but for small time scale of up to one hour. Reference [32] shows

similar results where the Auto Regressive (AR) method is used to predict only the next one-

hour and a small MAE is obtained.

A hybrid between the ARMA model and the ARCH (Auto Regressive Conditional

Heteroscedasticity) is used by [31] in which comparison is done with the classic Persistence

model. Due to volatility clustering, variance changes over time and the advantage of the

ARCH has been utilized. The results show high MAE of up to 29.37% though. However,

when the Wavelet Theory is combined with the neural network it shows very good results

[36]. The original wind speed is decomposed here during the wavelet theory and then the

16

output is fed to Levenberg–Marquardt Back Propagation (LMBP) neural networks. These

are built for the forecasting of every component respectively thus obtaining good results.

The WT with a forecasting method based on empirical mode decomposition (EMD) shows

4.53% MAPE [1]. Thus, this method may be reliable when combined with other methods

yielding favorable results.

The Grey Model is a time series model and is used by [37] to predict for the next 3 hours.

There is a little documented research and this paper is the only one presenting a pilot study

about it. The case study corresponds to a wind farm located in Penghu, Taiwan, to verify the

feasibility of Grey Model and satisfactory results are obtained.

2.3.3 Medium Term Forecasting

Medium term forecasting is the prediction between 6 hours to one day ahead. Most of the

methods developed for this time-period are based on ANN approaches, physical weather

models, and hybrid models combining both of these or some new techniques. These include

fuzzy logic, wavelet transform, ensemble predictions, and spatial correlation. These new

methods when combined with the existing approaches are expected to give promising

results.

Reference [3] uses a two-stage forecasting method up to 30 hours ahead. First the

decomposition of wind is performed and then ANN is applied to each signal. Then, predicted

wind speed is used to forecast wind power output through nonlinear input–output mapping

using Feed Forward Neural Network (FFNN). The RMSE obtained is 7.081% for 30 look-

ahead hour giving satisfactory results.

Support vector regression model is developed by [38] using three years data of wind speed,

temperature, barometric pressure, wind gust, wind direction, humidity on an hourly average

basis to predict hourly wind speed. It is observed that the MAPE is around 7% with a

correlation coefficient close to 1.

Spatial correlation is suggested by [30] in which local and spatial relations of the wind speed

are considered so as to improve the efficiency of short and long range forecasting. This

method is used to test the timeframe from a few minutes to several hours thus investigating

the method for a large timescale. Investigations are supported by measurements at several

17

terrains during a whole year. An Artificial Neural Network model is used to forecast wind

speeds and the associated electrical power for a few minutes up to a few hours ahead leading

to MAE 20- 40 % better than the persistent ones.

2.3.4 Long Term Forecasting

Long-term forecasting is prediction from 1 day to 1 week ahead or more. It is very important

in well-run electricity markets where long-term scenarios are taken into consideration.

A new type of physical method to predict probability density functions of wind power

generation for 1-to-10 days ahead by ‘weather ensemble predictions’ (WEP) is developed by

[39]. The model is calibrated and smoothened at five U.K. wind farm locations to accurately

predict the uncertainty in weather conditions. Comparison of results with the ARMA model

and Generalized Auto Regressive Conditional Heteroscedasticity (GARCH) show WEP

giving more accurate results over a period of a week. It is disappointing to see that beyond

two days ahead, this method is matched by the simplistic approach that uses the mean of the

actual wind power for the same day in each of the previous five years.

Reference [40] deals with long-term of up to 72 hours wind speed and power forecasting

based on meteorological information. Two novel and optimal schemes are suggested for the

update of the recurrent network’s weights based on the recursive prediction error algorithm.

Based on the data and network chosen an exhaustive set of experiments is carried out.

Results show that Recurrent Neural Network (RNNs) improve over persistence method by as

much as 50% and may still be improved if additional ‘node’ predictions are included. This is

done in [41] where RNN forecasts in lesser time and without too much data. They forecast

for each month separately and results are more accurate than Feed-Forward Neural Network

(FNNs) for several hours to day-ahead forecasts.

For non-Gaussian error distribution, decreasing the entropy minimizes the error [42].

Minimum Error Entropy (MEE), Maximum Co Entropy (MCE) or MEE with fiducially

points (MEEF) are decreased instead of decreasing the variance (known as MSE criterion)

and hence prove more suitable for training of forecast models. Tests based on MCE and

MEEF criteria against minimizing its variance (MSE) criterion were conducted on FNNs

that had one hidden layer of nine neurons for 72-hour ahead forecasts in half hour intervals.

This has immense scope when integrated into more powerful wind prediction systems.

18

Table 2.1: Time Scale and Error Classification for Different Technique

Time Horizon Range Methods MAPE Error Scope

Short-Term

Very

short

term- Few

seconds to

30

minutes

ahead

Persistence (P) 30% - Not many

comparison

papers available

-30 min ahead

forecast require

models still

EnKF 5.06%

ARMA 7% for 1 hour

ARIMA 18% for 6 hours

ANN <1%

Grey Predictor 27% better than P

Short

term- 30

minutes to

6 hours

ahead

EMD 4.53%

ARCH 29.37%

Grey Model Average is good

Medium-Term

6 hours to

1 day

ahead

FNN 12% better than P -Complex

Statistics

(Huge Scope)

Spatial

Correlation

40% better than P

Support Vector

Machine

7%

Long-term

1 day to 1

week

Or more

ahead

RNN Better than FNN -Non-Gaussian

Error

Distribution

Function and

Entropy Based

Criteria (New)

WEP Not mentioned

GARCH 27%

Table 2.1 summarizes wind speed prediction methods for different time scales. It shows the

classification of the main methods that fall under different time scales. Also, it shows the MAPE

error or the error comparison with the Persistence Model along with the scope of the methods for

respective time division.

19

2.3.5 Data Preprocessing

Wind speed forecasting is a very involved and specialized area. Generally, the forecast

method developed for one site or wind farm does not suit other due to a variety of reasons

like change in terrain, different wind speed patterns, different atmospheric factors such as

temperature, pressure, humidity. But ‘generalization capability’ of forecast model can be

improved by considering the design issues.

Generally, we feed data as input but it is sometimes directly fed into the forecast model such

as ANN, time series. Reference [43] shows that various differences are evaluated in the

output when these data are fed directly and are not preprocessed. Preprocessing means to

divide input so that a ‘local’ model may be designed for each subclass [44]. The standard

properties are extracted: standard deviation, mean, variance and slope from the input data.

The irregularities in data are filtered and the model becomes simpler.

2.3.6 Design and training of forecast models

Wind power forecasting models are studied and tested for one location and usually cannot be

applied to any other site. Developing a generalized, portable and easy-to-operate model is a

big challenge. Such a method is presented in [45] that has strong robustness and can be

easily modified for different wind farms with minimum effort. The results are 40% better

than the Persistence model and MAPE from 9.9% to 13.64% are reported.

Excessive training is not encouraged by some researchers and should be avoided or else

models tend to over-parameterize or over fit the input data. This applies to ANNs and time-

series models such as ARMA. This ends up with ‘poor’ forecasts in test data set. Reference

[44] shows that over fitting is a result of excessive complexity and lack of clear guidelines

on how many parameters in a model should be taken, for a given sample size. Many

researchers avoid this problem by reducing the weight of the number of neurons.

2.3.7 Selection of number of input nodes and output

Selection of the right numbers of input and output nodes is imperative because redundant

input data slows the process when including pressure and temperature [6]. To avoid this,

20

mostly inclusion of various weather variables is a prerequisite for accurate forecasts. Also, if

too old past values are used, they would be inappropriate to recent patterns, i.e., low auto-

correlation will weaken forecast functioning and should be avoided.

2.3.8 Implementation and Validation of forecast models

Implementation of a forecast model means to determine its parameters. Selecting optimal

number of model parameters is challenging and techniques like trial and error method, and

Box method [44] have been used widely for models using ARCH, ANN or fuzzy system

techniques. This is especially difficult for ANNs or fuzzy systems, as selecting good design

and electing proper rule base is vital otherwise the results are affected.

2.3.9 Persistence Method

Several methods are used for wind forecasting but the Persistence model is the benchmark

method. The simplest methods are based on climatology or averages of past production

values. The most popular and simplest form to forecast the wind is the Persistence method. It

is also known as ‘Naïve Predictor’. It uses the simple assumption that the wind speed at time

t+x is the same as it was at time t. In other words it states that the future wind values will be

the same as the last measurement. Unbelievably, it is more accurate than most of the

physical and statistical methods for very short to short-term forecasts. Therefore the MAPE

of each method is obtained and compared with the Persistence method. But as expected the

accuracy lessens with increasing time [2, 11]

A case study from Tasmania, Australia is shown in [46] for very- short term forecasting.

They use the ANFIS to forecast wind vectors rather than wind speed or power. In this paper

more importance is given to wind direction rather than wind speed. The model is trained for

21 month time series in steps of 2.5 minutes. Results show that the ANFIS produced less

than 4% MAPE compared to approximately 30% for the Persistence model.

2.3.10 Physical Approach

Physical systems use parameterizations based on a thorough physical depiction of the

atmosphere. Usually, at the location of the wind farm, wind speed given by the

meteorological experts is downscaled to the onsite situations.

21

Several physical models have been developed based on using weather data and the

topography for wind speed and wind power forecasts. Numeric Weather Prediction (NWP)

models solve complex mathematical equations using weather data such as temperature,

pressure, surface roughness, hindrance, effects of orography, scaling of the local wind speed

within wind farms and their layouts and wind turbines power curves. NWPs require super

computers, as they are highly computationally demanding. NWPs run at most 1 or 2 times a

day. Therefore, these are used for up to 6 hour ahead forecasts and show better results when

weather conditions are stable. NWPs include global forecasting systems [41] or HIRLAM

(High Resolution Limited Area Model) [47] for instance.

These methods provide wind speed estimates for a grid of adjacent points around the wind

generators. These forecasts use a meso-scale or micro-scale model for the downscaling,

which interpolate the level of wind speed forecasts to that of the wind generators [48]. A

detailed description of the environment surrounding the wind generators is essential to run

the downscaling models. This is one of the main difficulties in the implementation of

physical models.

Reference [47] uses synthetic aperture radar (SAR) and medium resolution imaging

spectrometer (MERIS) onboard the environmental satellite (ENVISAT) in synergy to

analyze severe weather systems. Maximum wind speeds of up to 25 m/s were measured by

SAR and confirmed by the models. Significant differences were observed in the location of

the maxima. HIRLAM showed differences in meso-scale turbulent behavior and coastal

shadowing. Meso-scale features, e.g., downburst due to cloud patterns with a diameter of up

to 15 km, were not detected by HIRLAM.

2.3.11 Statistical Approach (ANN and Time series models)

This approach includes two main types namely ANN and Time series models. They train the

measured data and then use the difference between the predicted and the actual wind speeds

in immediate past to tune the model parameters [49]. It is easy to model, inexpensive, and

provides timely predictions. It is not based on any predefined mathematical model and is

rather based on patterns. Errors are minimized when patterns match with historical ones.

Sub- classification of this approach is: time-series based models and neural network (NN)

based methods.

22

Some of the most popular techniques include ARMA models, ARIMA, seasonal- and

fractional-ARIMA, grey predictors, linear predictions and exponential smoothing. The

ANNs may also be trained using past data taken over a long time frame to learn the

relationship between input data and output wind-speeds. ANNs have an input layer, where

historical data are fed for learning, hidden layer(s), and an output layer providing forecast

results. Models such as FNNs, multi-layer perceptron (MLP), RNNs, radial basis function

(RBf) NNs, Adaline networks are all examples of the ANN models.

The method by [50] uses a new statistical approach that combines artificial intelligence and

fuzzy logic techniques. It provides a preliminary forecasting of wind power based on Radial

Basis Function (RBf) network that enhances its performance and estimates the quality of the

forecasts using fuzzy rules. Results obtained with the proposed method for an actual wind

farm show the applicability of the method and the improvement over other methods.

A wind power forecasting is carried out in two stages by [3]. First, a multi-resolution

analysis-based forecasting using Adaptive Wavelet Neural Network (AWNN) is developed

and then a feed-forward neural network (FFNN) is used for nonlinear mapping between

wind speed and wind power output for predicting wind speed up to 30 hour ahead. It shows

that with the discontinuous wind power series FFNN is better suited than AWNN.

Other statistical methods give comparatively efficient forecasting models that do not require

any data beyond historical wind power generation data. However, the accuracy of the

prediction from these models decreases remarkably as the time horizon is increased.

Reference [51] uses stochastic simulation models. They remove the annual and daily

periodicities of the measured data and modeled transformed hourly average wind speeds,

taking into account autocorrelation, non-Gaussian distribution and diurnal non-stationary

and fit an ARMA process to wind speed data. It was observed that this model would be

useful only for a short time range and not beyond two hours. Reference [52] uses an ARMA

model to forecast hourly average wind speeds for five sites in Navarra, Northern Spain. They

used site and month specific parameters for the ARMA model. The ARMA model usually

outperformed the persistence method for the 1-hour forecast, and was better in RMSE and

MAE for higher horizons up to 5 hours ahead.

Reference [53] found an ARMA (p, q) process suitable for both wind speed simulation and

23

forecasting. The inclusion of the diurnal variation was deemed important since the (mainly

thermally driven) climate of Pakistan exhibited quite strong uniformity especially in the

summer months. Errors have not been calculated and only comparisons of actual and

predicted data values have been shown in this paper. Reference [54] uses the differences of

wind speeds from the moving averages (differenced pattern method) and found this

technique to be superior to the wind speed normally used as input. They achieved

improvements of up to 13% over the persistence method, while for the same time series the

standard neural network approach yielded only 9.5% improvement.

Reference [55] found that wind power and especially wind power variability from large

offshore wind farms (Horns Rev and Nysted) occur in certain regimes, and therefore tested

“regime-switching approaches relying on observable (i.e. based on recent wind power

production) or non-observable (i.e. a hidden Markov chain) regime sequences” for a one-step

forecast of 1-min, 5-min and 10-min power data. It is shown that the regime-switching

approach outperforms those based on observable regime sequences. The reduction in one-

step ahead RMSE ranges from 19% to 32% depending on the wind farm and time resolution

considered.

2.3.12 Hybrid Approach

Combination of different approaches such as short-term and medium-term models, physical

and statistical approaches or mixing alternative statistical models are referred to as a hybrid

approach. The aim of hybrid models is to benefit from the advantages of each model and

obtain a globally optimal forecasting performance.

For example, a semi empirical model is developed that recovers wind direction and improves

wind speeds from Special Sensor Microwave/Imager (SSM/I) measurements. It uses

radiative transfer and ANN techniques combined with Special Sensor Microwave/Imager

(SSM/I) to get ocean surface wind speeds and direction [5]. Results show that this

combination can considerably improve the impact of these data in NWPs compared to using

SSM/I.

Some techniques have become very popular in recent years, for instance the Spatial

Correlation, Fuzzy logic, Wavelet transform, Ensemble Predictions and so on. Reference

[30] uses the spatial correlation of wind speeds. The wind speed time-series of local point

24

and its neighboring sites are employed to predict the wind speed by ANNs or adaptive

neuro-fuzzy networks. Only historical data of the site under consideration is used, and no

values related to the physical phenomena, such as pressure profiles, terrain shape or wind

speed at different sites. This is probably the reason why their accuracy is limited.

Reference [6] presents a fuzzy model with two premise variables–the wind speed and its

direction. It uses a genetic algorithm to determine model parameters. It trains data measured

at various stations in and around the wind park. Thus, the autocorrelation and cross-

correlation between local and remote wind speed time series are exploited to improve the

error from a few minutes to several hours ahead.

Table 2.2 classifies different types of wind speed forecasting methods. It further categorizes

them into different kinds. It also gives some common examples that have been studied in this

research and provides a few remarks.

The following hybrid techniques have been researched in this thesis and are reviewed as

follows:

2.3.12.1 ANN-EnKF Hybrid

This section covers two major areas of predicting wind speed. They are Artificial Neural

Networks and Kalman Filter. Combination of ANN and EnKF acts as an output correction

scheme. With observations, the EnKF will correct the output of an ANN to find the best

estimate of the wind speed.

Artificial Neural Networks (ANN) have been a good selection to model and forecast time

series. ANN models can represent a complex nonlinear relationship and extract the

dependence between variables through the training and learning process. They are simpler to

construct and require shorter development time. One just needs to identify correct inputs, set

up the network structure and then model a training algorithm, which gives the best prediction

25

Table 2.2: Basic Wind Forecasting Methods

Forecasting Methods Subclass Examples Remarks

Persistence Methods /

Naive Predictor

-

P (t+k) = P (t)

- Benchmark approach

-Very Accurate for very

Short and short term.

Physical Approach

Numeric

Weather

Predictor

(NWP)

- Global Forecasting

System

-HIRLAM, etc.

-Use of meteorological

data such as wind speed

and direction, pressure,

temperature, humidity,

terrain structure etc.

- Accurate for long term

Statistical Approach

Artificial

Neural

Network

(ANN)

- FNN

-MLP

-AWNN

- Radial Basis Function

-ADALINE, etc.

- Accurate for short

term

-Their Hybrid structures

are useful for medium

to long term forecasts

- Usually, outperform

time-series models

Time-series

Models

- ARMA

- ARIMA

- Grey Predictors

- Linear Predictions

-Exponential

Smoothing, etc.

- Accurate for short

term

- Some very good time-

series models supersede

ANN structures

Some New

Techniques

-

- Spatial Correlation

-Fuzzy logic

- Wavelet transform

- Ensemble Predictions,

etc.

- Spatial Correlation is

good for short term

- Entropy based training

of models improves

performance

Hybrid Structures

_

- Radiative Transfer

+ANN

-Spatial Correlation +

ANN

-Fuzzy+ Genetic

Algorithm, etc.

- Radiative Transfer

+ANN Structure are

very accurate for

medium and long-term

forecasts.

26

results. References [56-58] proposed different wind prediction methods using ANN models.

They perform a non-linear mapping between inputs and outputs and provide an alternative

approach for wind prediction. References [59-60] showed genetic algorithm ANN that reduce

workload, improve efficiency, and also improve forecasting accuracy

Sequential methods such as optimal interpolation and Kalman Filter belong to the second

family of methods. In these methods, observations are used as soon as they are available to

correct the present state of the model. The KF is an excellent technique for data assimilation

as it takes maximum advantage of observations. On the other hand this method has a few

practical problems when it is used in operational applications for large non-linear models. In

these situations the computation of the forecast error covariance matrix and Kalman gain can

become enormous job both in computer power and computer storage.

A Kalman filter is a linear estimator. It is used to estimate the state of a linear dynamic

system by using measurements linearly related to the state of the system but corrupted with

noise. A Kalman filter is a recursive data processing algorithm. It is a tool that does not

require all previous data to be kept in memory. Finally this type of filter is optimal: it

calculates the best possible estimate (minimum variance) for the state of the system. This

research uses the Ensemble Kalman Filter for non-linear systems. References [61-62] used

EnKF and the results gave a 10-30% reduction in RMS error. Reference [63] showed that

High Order Neural Network (HONN) trained with an extended Kalman filter gave absolute

errors to be very low around 1.23 %. References [64-65] had a new wind estimation method

in which a Kalman filter was used to produce high quality wind estimate.

A lot of research has been done on the various methods individually and also in hybrid form

but the combination of ANN and Kalman Filter remains an unexplored area of research. This

research is an attempt to come out with a new technique based on the hybrid of these two

techniques.

2.3.12.2 ARMA-WT hybrid

Wind speed forecasting is a very involved task and generally the forecast method developed

for one site or wind farm does not suit the other due to a variety of reasons like change in

terrain, and different wind speed patterns just to name a few. A lot of research is being done

27

in order to decrease the forecasting error but there is still scope to develop methods for short

term forecasting.

The Ensemble Kalman Filter (EnKF) method is explained in [35]. Reference [36] uses EnKF

to predict 10 min ahead wind speed that gives an error of about 5.06%. Linear prediction

model of the ARMA is shown in [34] where results from 1s to 5s only are reported. A hybrid

between the ARMA model and the ARCH (Auto Regressive Conditional Heteroscedasticity)

is used in [31] where comparisons are drawn with the classic Persistence model. The results

however show high MAE (Mean Absolute Error), up to 29.37%. The mean absolute error

(MAE) is a quantity used to measure how closely the forecasts or predictions follow to the

eventual outcomes. When the Wavelet Theory is combined with the neural network it shows

satisfactory results [36]. Reference [1] shows a similar result when the WT is combined with

an empirical mode decomposition (EMD) showing 4.53% Mean Absolute Prediction Error

(MAPE).

The Grey Model is a time series model and is used in [37] to predict the next 3 hours. Since

there is very little research about this method, a pilot study has been done in this paper,

which gives significant forecast errors. Developing a generalized, portable and easy-to-

operate model is therefore a big challenge. Such a method is presented in [45] that has strong

robustness giving 40% better results than the Persistence model.

Reference [44] shows that over fitting is a result of excessive complexity and lack of clear

guidelines for the number of parameters to be chosen while modelling a given sample size.

A case study from Tasmania, Australia is shown in [46] for very-short term forecasting in

which more importance is given to wind direction rather than wind speed. Numeric Weather

Prediction (NWP) models are used for up to 6 hour ahead forecast and show better results

when weather conditions are stable. NWPs include global forecasting systems [41] or High

Resolution Limited Area Model (HIRLAM) [47] for instance. The method in [50] uses a

new statistical approach that combines artificial intelligence and fuzzy logic techniques.

Reference [66] shows different approaches of ARMA for short term forecasting of wind

speed. The effectiveness of ARMA-GARCH is reported in [67]. It also demonstrates

different GARCH approaches consistently improving the modelling sufficiency of the mean

28

wind speed. However, the output prediction decreases with increase in the height, which is a

drawback.

Reference [68] shows the impact of wind speed forecasting on the dynamic response of wind

turbine. It calculates rotor speed and aerodynamic torque to estimate the effectiveness of

wind speed by an inversion of a static aerodynamic model and hence shows that this may

also be helpful for wind turbine response. Wind speed prediction is not only a problem for

wind power prediction but also for aqueducts in China that still withstand wind loads

constantly [69]. These wind loads shorten aqueducts fatigue life along with large amplitude

sloshing of water and further leading to structural resonance. Another application can be

seen in [70] where variable-speed wind turbine generators (WTGs) employ anemometers to

measure wind speed so that the desired optimal shaft speed can be derived. Results show that

the wind speed was accurately estimated and hence proved useful.

A lot of research has been done to forecast wind speed but still work needs to be done to

develop methods to realize robust and more accurate predictions. Some hybrid methods have

gotten satisfactory results and smaller MAPE but they have not been tested for more than

one-hour prediction. The ARMA model on the other hand has shown good results for long

term forecasting. In this thesis, ARMA is combined with the WT, given the complementary

strengths of both techniques, to assess their forecasting accuracy.

2.3.12.3 ARMA-ANN-AIC hybrid

Wind speed forecasting is a very active and specialized research and development area.

Generally, the forecast method developed for one site or wind farm does not suit another due

to several factors, e.g., change in terrain and different wind speed patterns. Extensive

research has been reported to decrease error but there is still a lot of scope to improve the

short term forecasting techniques.

Reference [71] shows a GA fitness value relying on the deviation between the actual plant

output, with or without an additive noise, and the estimated plant output. Simulation results

show in detail the efficiency of the proposed approach however it does not directly tell us the

best order. Reference [72] combines the effectiveness of the Multi Model Partitioning theory

29

with the robustness of Evolutionary Algorithms but gives a bit complicated approach with a

10% error.

Reference [73] uses Artificial Neural network (ANN) technique and compares it with some

known and widely acceptable techniques. The comparisons are entirely based on the value of

the coefficients obtained. The results show that the use of ANN also gives an accurate

computation of the coefficients of an ARMA system. But it only stops to the determination

of the parameters and not the true order.

Reference [74] shows that the ARMA model based on System Identification Toolbox of

MATLAB is valid to forecast wind signal and can reflect the future characteristics of the

signal. The average relative error of the model is found to be 6.9%, which can be drastically

reduced. Reference [46] uses ANN but fails to give any explanation on the analysis of the

algorithm used. Reference [75] proposes to utilize ANN for order identification that uses

Extended Sample Auto Correlation Function (ESACF) method to determine the order of

ARMA. However, the proposed method finds it hard to estimate the proper order when the

p and q values become too great. In addition, it takes a long learning time. The present work

proposes to combine ANN with AIC and the simulation studies are conducted to show that

the proposed hybrid method gives better satisfactory results. The gaps in these papers are

filled with the proposed hybrid.

There are several publications that report to predict ARMA and its true order but the

combination of ANN- Akaike Information Criteria has not been used for order estimation.

Therefore, in this research the ARMA model is combined with the ANN and AIC to give

good and accurate results.

2.3.12.4 ORPD-PSO-GT hybrid solution for the Economic Dispatch

problem

In the last decades, many useful studies [76-79] based on classical techniques for solving

ORPD problems have been carried out. These include Newton and quadratic techniques

nonlinear programming (NLP), successive linear programming, and mixed integer

30

programming just to name a few. These approaches are broadly categorized as constrained

optimization techniques. There are still a number of issues to be addressed with regarding

practical power systems. One is when a local minimum is returned instead of a unique global

minimum or the other is the inherent integer nature of the problem among the few crucial

problems. This means that most control devices have pre-specified discrete values. Thus

irrespective of the accuracy of the continuous solution, it is impossible, without making

some reasonable approximations, to assign these values directly to the physical control

devices.

A lot of attention has been given to the computational intelligence-based, heuristics methods

over the years such as GA [80], PSO [81] and differential evolution (DE) [82] since they

have been giving better results as compared to the classical methods [83]. GA’s are

stochastic search techniques based on the mechanism of natural selection and survival of the

fittest. Reference [84] shows how to correct abnormal bus voltages to prescribed limits for

the Nigerian Grid. Reference [85] describes the basics of PSO, different kinds, and few

hybrids including GA-PSO techniques, and its applications in power systems. It shows that

the PSO algorithm is one of swarm intelligence techniques based on simulating the food-

searching behavior of birds, and has been widely used in power systems. These techniques

have shown effectiveness in overcoming the disadvantages of classical algorithms such as

slow speed since they lack the potential to finding good solutions. To the contrary, PSO and

DE have received great attention from researchers because of their speed, novelty and

searching power.

Reference [86] presents a PSO for reactive power and voltage control (Volt/Var Control)

considering voltage security assessment. It shows that PSO only requires less than 50

iterations for obtaining sub-optimal solutions even for large-scale systems. Reference [87]

explains PSO as a tool for loss reduction of optimal power flow (OPF) problem. Reference

[88] compares three new PSO with the state of the art PSO algorithms for the optimal

steady-state performance of power systems, namely, the reactive power and voltage control.

It proves Coordinated Aggregation based PSO to be better over others. Reference [89] shows

a modified particle swarm optimization (MPSO) method to realize ORPD considering

voltage stability improvement. It shows decrease in system loss by 11.4%. Although these

31

papers have considered PSO as a good technique there is no research that uses this in

conjunction with the GT.

DE, however, is not completely free from the problems of slow and/or premature

convergence as shown in [90]. It describes a family of improved variants of the DE/target-to-

best/1/bin scheme, which utilizes the concept of the neighborhood of each population

member and tries to overcome the disadvantages of becoming stagnant. Reference [91] tries

to overcome the problem of voltage instability of ORPD by employing a new algorithm

named shuffled frog leaping algorithm (SFLA). It calculates active power losses to be very

low. Reference [92] compares the evolutionary programming (EP), PSO, DE and the

proposed hybrid differential evolution (HDE) algorithms. The HDE proves to be the best for

calculation of fuel cost. Reference [93] explains the reactive power dispatch problem in

detail and compares the solution with PSO and DE algorithms. Hybrid multi swarm PSO

gives the minimum transmission losses.

Reference [94] presents a solution to the reactive power dispatch problem with a novel

particle swarm optimization approach based on multi agent systems (MAPSO). An agent in

MAPSO represents a particle to PSO and a candidate solution to the optimization problem.

The computational time taken by MAPSO is found to be very high where almost 70% of the

total computing time is spent on the load flow algorithm. Reference [95] proposes different

HDE algorithms, which tend to take a larger average convergence time than the single

optimization scheme since two sub-optimization processes are executed in a single

simulation run.

Several papers have used different algorithms to come out with a solution of ORPD but they

seem to be complex. Reference [96] uses a seeker optimization algorithm (SOA)-based

reactive power dispatch method, which is a novel heuristic stochastic optimization

algorithm, based on simulating the act of human searching. It shows that the basic form of

the proposed SOA algorithm can only handle continuous variables. Reference [97] proposes

hybrid multi-swarm particle swarm optimization (HMPSO) algorithm to solve the ORPD

problem where only the control variables are adjusted and compared with different

algorithms to make the solution better. It comes out to be more complex as compared with

PSO and DE algorithm and requires more computing time.

32

Some researchers have been working on GT in power systems to identify faults in the

electric system. However, there is not much that has been done in wind generation and

dispatch. Reference [98] shows how GT can be used to number different nodes and how

positive direction of current should be taken. Reference [99] introduces tie-set GT that uses

loops to realize distributed control in local units. It shows the application of GT to smart grid

by breaking the graph into various required and not required sections. Reference [100]

develops a polynomial time algorithm to decompose a graph but this paper lacks the means

to identify the elements of a set and also the order of the largest graph in the set. Reference

[101] presents an electrical network graph partitioning technique that divides power

networks into zones and then tests it on the IEEE 39-bus and IEEE 118-bus. It shows the

capability of GT for model reduction of complex power systems.

2.4 Summary

A lot of research has been done to forecast wind speed but still work needs to be done to

develop methods to realize robust and more accurate predictions. Some hybrid methods have

got satisfactory results and smaller MAPE but they have not been applied for more than one-

hour prediction. The ARMA model on the other hand has shown good results for long term

forecasting. Also, results from 1 hour ahead to 50 hours ahead can be obtained accurately.

The WT is chosen for this combination since wavelets allow complex information such as

music, speech, images and patterns to be decomposed into elementary forms at different

positions and scales and subsequently reconstructed with high precision. Therefore, ARMA

is combined with the WT, given the complementary strengths of both techniques, to assess

their forecasting accuracy

Moreover, there has been research on the various forecasting methods but the combination of

ANN and Kalman Filter remains an unexplored area of research. This research has explored

this hybrid to come out with a new technique. ANNs are simpler to construct and require

shorter development time whereas KF is an excellent technique for data assimilation as it

takes maximum advantage of observations.

33

A number of factors determine the economics of utility-scale wind energy and its

competitiveness in the energy marketplace. It includes the cost of wind energy that varies

widely depending upon the wind speed at a given project site, improvements in turbine

design bring down costs, optimal configuration of the turbines, cost of financing affects,

transmission, tax, environmental, and other policies also affect the economics of wind. The

potential economic benefits arise in the system-wide generation cost savings and in the

ancillary service cost savings. The results from the new wind-forecasting model may be

implemented into a power system economic dispatch model, which may take into account

both spatial and temporal wind speed correlations. This, in turn, will lead to an overall more

cost-effective scheduling of system-wide wind generation portfolio. Reference [102] shows a

model to study how changes in the trading arrangement or changing the imbalance pricing

system would affect different players in the electricity market. Reference [103] analyzes

different market scenarios and has estimated to decrease MAPE to 3.7% for wind power

forecast until 2020.

A lot of work still needs to be done to develop methods to realize robust and more accurate

solutions. This research proposes PSO as a solution of OPRD using GT to analyse the

relationship between the forecast wind speed and the ED problem. Some hybrid methods

have got satisfactory results and small transmission losses but they have not been tested

outside their range. The PSO technique can generate high-quality solutions within shorter

calculation time and have more stable convergence characteristic than other stochastic

methods. Although the PSO seems to be sensitive to the tuning of some weights or

parameters, research is still in progress to prove its potential in solving complex power

system problems.

34

Chapter 3

ANN-EnKF Hybrid

3.1 Introduction

This Chapter focuses on the ANN-EnKF hybrid. A surrogate of ANN has been built and has

been shown with the help of a schematic diagram. Also, EnKF has been discussed and the

mathematical equations are shown in the next section. The reason for choosing ANN and

EnKF has been justified in Section 2.6. ANN has been combined with EnKF with the help of

a mathematical matrix equation to come out with a hybrid to act as an output correction

scheme. This hybrid has been tested on MATLAB for Auckland, New Zealand for 3 hourly

and 6 hourly data. This hybrid proves to be beneficial for this short term forecasting and

hence, this research proves to be useful for wind speed forecasting.

3.2 Surrogate of ANN

Surrogate models are approximate models that copy the behavior of large scale models to

small scale models that makes it cost efficient. It is constructed using data driven models and

choosing appropriate data points placed together. ANN maps the input vector into

corresponding output vector while other values need not be known makes it imperative. This

makes ANN very useful to mimic nonlinear relationships without the need of any already

existing models [60].

Reference [104] showed a dynamic system in which the state at time k depends on the system

35

state for preceding time state and wind speed. Therefore, a recurrent model is designed to

develop the desired state form. In this paper the following time lagged ANN surrogate is used

to map the wind model in reduced space in which FNN is the surrogate model.

𝑋𝑆�� (𝑘 + 1) = 𝐹𝑁𝑁[��𝑆

𝐻 (𝑘), ��𝑆𝐻(𝑘 − 1),… ��𝑆

𝐻 (𝑘 − 𝜏𝐻), 𝑈𝑆 (𝑘), 𝑈𝑆 (𝑘 −

1),… , 𝑈𝑆 (𝑘 − 𝜏𝑈)] (3.1)

𝑋𝑆�� (𝑘 + 1) is the predicted output of the model. So this will give one-step ahead output. 𝜏𝑈

and 𝜏𝐻 are the time index parameters.

Fig 3.1. Schematic presentation of ANN surrogate

36

Figure 3.1 shows the dynamic recurrent network with feedback output to input in a schematic

diagram. The output 𝑋𝑆�� (𝑘 + 1) is fed back through unit delay operator Z-1 to the input that

consists of past system states.

ANN is a model approximate i.e., it is a model of a model. Its accuracy is lower than the

original but can be increased by clustering of data. Being a data driven model, this model is

not sensitive to type of wave model and can be set up by using the data or other generation

models.

3.3 Ensemble Kalman Filter

The Kalman filter tries to estimate the state x that belongs to kn of a discrete-time controlled

process that is governed by a linear stochastic differential equation. The recursive form of a

Kalman filter that is suitable for problems with a large number of variables is referred to as an

ensemble Kalman filter or EnKF.

Now, our process again has a state vector, but that process is now governed by the non-linear

stochastic differential equation model M which is

𝑋𝑓(𝑘) = 𝑀 [ 𝑋𝑓(𝑘 − 1),𝑈(𝑘 − 1)] + 𝑤(𝑘 − 1)] (3.2)

𝑦°(𝑘) = 𝐻 (𝑘)𝑥𝑓 (𝑘) + 𝑣 (𝑘) (3.3)

where xf(k) denotes forecast at time k, U(k) is forcing of the system and M is one time step of

the model. Reference [29] has used these equations to study the Kalman filter.

In practice, the process noise covariance and measurement noise covariance matrices are

given as:

𝑝(𝑤)~𝑁(𝑂, 𝑄)

𝑝(𝑣)~𝑁(𝑂, 𝑅) (3.4)

37

In order to obtain an optimal estimate, it is needed to combine the measurement taken from

the actual system modeled by Eq. (3.2) with the information given by the system model (Eq.

(3.3)). The forecast state at time k, denoted by xf, is the forecast from observation time k-1 to

observation time k. It is given by

𝑥𝑓(𝑘) = 𝑀[𝑥𝑎(𝑘 − 1),𝑈(𝑘 − 1)] (3.5)

where xa(k-1) is the analyzed system state. Observation y(k) is updated by

𝑥𝑎(𝑘) = 𝑥𝑓(𝑘) + 𝐾[𝑦°(𝑘) − 𝐻(𝑘)𝑥𝑓(𝑘)] (3.6)

Also, the Kalman gain is given by

𝐾(𝑘) = 𝑃𝑓(𝑘)𝐻(𝑘𝑇)[𝐻(𝑘)𝑃𝑓(𝑘)𝐻(𝑘𝑇) + 𝑅]−1 (3.7)

It is the minimum variance gain and Pf(k) is the forecast error covariance matrix. This

covariance can be calculated by a finitely number of randomly generated system states [105].

3.4 Hybridizing ANN Surrogate and Kalman Filter

In the combined method, the output of the ANN will be considered as state vectors. The state

vectors are provided to the EnKF. By help of the observations, the EnKF will correct the

output of the ANN to determine the best estimate of the system (or analyzed state). The

analyzed state will return to the inputs of the ANN for the next time step. There is a

difference between inputs of network, which come from forcing or from feedback loop. To

deal with the time delayed states in Eq. (3.1), the Kalman filter formulation proposed by [106]

is employed and an extended vector is taken as follows

𝑋𝑘[𝑋𝑠𝐻,𝑘 , 𝑋𝑠

𝐻,𝑘−1, … . , 𝑋𝑠𝐻,𝑘−𝜏𝐻]𝑇 (3.8)

This new state vector expands the dynamic equation to the following equation

38

[ 𝑋𝑠

𝐻,𝑘+1

𝑋𝑠𝐻,𝑘

𝑋𝑠𝐻,𝑘−1

⋮

𝑋𝑠𝐻,𝑘−𝜏𝐻+1]

=

[

𝐹𝑁𝑁(… )

10⋮0

01⋮0

……⋮…

0 00⋮1

0⋮0]

[ 𝑋𝑠

𝐻,𝑘

𝑋𝑠𝐻,𝑘−1

⋮

𝑋𝑠𝐻,𝑘−𝜏𝐻]

+

[ 𝑊𝑠

𝑘

00⋮0 ]

(3.9)

FNN is given in Eq. 3.1 and I is the identity matrix. The compact equation of this is in

accordance with the general equation of Kalman filter and is

𝑋𝑘+1 = 𝑀𝑠[𝑋𝑘] + 𝑊𝑘

𝑊𝑘 = [𝑊𝑠𝑘, 𝑂, 𝑂, …𝑂]𝑇 (3.10)

where Wk is the extended noise vector and Ms is the new dynamic operator.

The observation Eq. 3.3 can be replaced by

𝑦°(𝑘) = 𝐻(𝑘)𝜋𝑟2𝑇 𝑋𝑠

𝐻,𝑘 + 𝑣𝑠 (3.11)

where vs(k) is the measurement noise in reduced space. Reference [106] gives derivation of

the measurement noise in the reduced space

𝑣𝑠(𝑡𝑘) ≈ 𝐻(𝑡𝑘)(𝑋𝐻,𝑘 − 𝜋𝑟2

𝐻 (𝜋𝑟2𝐻 )𝑇𝑋𝐻,𝑘 + 𝜇𝐻) + 𝑣(𝑡𝑘) (3.12)

This corrects the underlying model.

3.5 Results and Discussion

In the present study, wind data and the input-output patterns for training of a network are

taken from Auckland City, New Zealand for the year 2011 on a minute basis. The platform

chosen to test the available set of data is MATLAB. The six hourly wind fields are extracted

from the data from the period of January 2011 to July 2011, which means that data for 6

hours was extracted from each day’s data. For this, the total number of data values selected

39

were 1000. Wind speed data values are at an interval of 1 minute. These results are a proxy

for true wave states. Levenberg-Marquardt back propagation algorithm is used for this

analysis.

Since it is a comparison between the individual and the hybrid of the two methodologies,

different figures have been plotted for different ensembles in MATLAB. The six hourly data

were selected for the execution of wind speed model. The total number of data values taken

was 1000. The wind speed results are a proxy of the true wind states. The forward run time

is approximately 180s. The original wind data is shown in Figure 3.2. It shows the time

index for 1000 data values and the general pattern of wind speed in that location for the

particular period of time. The larger variances show large-scale patterns with slow changing

amplitudes representing stability whereas the smaller variances show small-scale patterns

with fast changing amplitudes representing instability.

Fig. 3.2 Original Wind Data

40

The first and foremost step in the prediction process is data normalization. This step is done

by transforming the actual wind speed data X into Normal wind speed data, Xn (i.e., Xn is a

normalized Gaussian Random Variable with zero mean and unit variance). The shape of the

Normal signal Xn is shifted down with negative values compared to the actual signal X due

to the normalization process as done in [126]. It is noted, negative wind speed is observed

throughout this thesis. This is because some wind speed is negative when the wind speed is

quite slow as in [127]. Therefore, the raw data may be dealt with before considering the

sampling frequency is not high if necessary.

There are two groups of data taken to prepare the surrogate model. The first one contains

1000 data values for training of network whereas the second group contains 100 data values

for testing of network. Levenberg-Marquardt back propagation algorithm is used for this

training of network. Initially 20 numbers of hidden neurons were taken to check the

performance and were increased up to a number where significant changes were not

observed. It was seen that with 53 hidden neurons there was no requirement of any further

improvement.

Fig 3.3 Output of ANN fed with original wind data

Figure 3.3 shows the output of ANN when fed with the original wind data. The first graph in

41

blue shows the original wind data. When it is fed to ANN, the second graph in pink is

obtained. The bottom graph in the following figure shows the ANN output, and the top graph

shows the original wind data. Data has been normalized here as done in [126].

The test data includes 100 data values that are not included in the training of network. In

spite of well-trained data the surrogate alone could not predict the exact wind speed in the

test period. This surrogate has both simulation and reconstruction errors.

As a solution, a hybrid of the surrogate with the Ensemble Kalman Filter is combined to

reduce the error. As this is a comparison of the individual and the hybrid techniques the

graph for the Ensemble Kalman filter is plotted which is shown in Figure 3.4. Three models

are plotted that include the reduced space model (red), which is used because of the

surrogate model. This field is projected into the reduced space and it can be re-embedded to

full space to reconstruct the original wave field. The wind model (green) is the one that is

depicted in such a way so as to see the variation of the wind model. The surrogate model of

ANN (blue) is the approximate model that copies the behavior of large-scale models to

small-scale models. These models are plotted for the Ensemble Kalman Filter. The effects

can be noticed when the same models are plotted for the ANN-EnKF hybrid. This shows the

much-improved result when the surrogate is coupled with the Kalman Filter. It gives much

finer and accurate prediction for the surrogate model.

This estimates the original wind speed to the closest prediction and proves to be beneficial

for the forecasting of the wind speed.

The same three models are now plotted for the ANN-EnKF hybrid and effect can be seen in

Figure 3.5. The ensemble size is chosen to be 100. It showed decreasing error with

increasing number of ensembles. The results are much improved when the surrogate of ANN

is coupled with the Ensemble Kalman Filter. It gives much finer and accurate prediction for

the surrogate model. This estimates the original wind speed to the closest prediction and

proves to be beneficial for the forecasting of the wind speed.

42

Fig. 3.4 Output of Ensemble Kalman Filter

The results show the MAPE to be 6.57%. The difference between the actual wind speed and

the one obtained from the hybrid is shown in Figure 3.6. The error is minimum thereby

confirming that this hybrid acts as an output correction scheme. There is not much difference

in the actual data gathered and the one obtained from the graph that proves the accuracy.

43

Fig. 3.5 Output for the hybrid of ANN and EnKF.

Fig. 3.6 Error Graph for ANN-EnKF Hybrid

44

The root mean square errors of the surrogate alone and the reconstructed hybrid are also

calculated and the results are shown in Figure 3.7.

Fig. 3.7 Root Mean Square Error for various wind speeds.

Table 3.1. Root mean square errors for different wave ensembles.

Ensemble Members Root Mean Square Error (m/s)

5 1.387

50 0.126

100 0.110

Table 3.1 shows that with the increasing number of ensembles the root mean square error

decreases. Hence, the underlying model acts as an output correction scheme.

45

3.6 Summary

This chapter presents the critical assessment of a unique combination of ANN and EnKF

methodologies. The mathematical equations of both the individual methods and the

combination have been presented. The proposed hybrid technique was applied on three

hourly data and six hourly data in MATLAB. First, the original wind data were shown.

Second, the changes from ANN and EnKF methods individually to the hybrid were shown

with different graphs. Finally, the RMS errors were calculated for different number of

ensemble sizes. The error decreases as the ensemble size increases. Therefore, the proposed

hybrid technique proves to be effective for very short term wind speed forecasting.

46

Chapter 4

WT-ARMA Hybrid

4.1 Introduction

In this Chapter, the Wavelet Transform (WT) along with the ARMA is chosen to form a

hybrid. These methods have been preferred since the proposed hybrid has a very high

potential for producing more accurate forecasting results. The Wavelet analysis is a novel

technique that is being adapted in various fields for forecasting purposes. The second

method used is the ARMA model, a time series model. The MAPE of the ARMA has been

proved to be between 7% for 1-hour up to 18% for the next six hours [11]. In this study,

these two methods have been combined to form a hybrid.

The ARMA model has been chosen in this research because it is suitable for high frequency

data. They produce accurate forecasts based on the historical patterns of the time series data.

They belong to the class of linear models and can represent both stationary and non-

stationary data. They do not involve the dependent variable; instead they make use of

information in the series to generate the series itself.

The ARMA model when compared to other conventional models is more robust in terms of

forecast accuracy as it takes seasonality into consideration. If the original series do not

exhibit season, non-seasonal ARMA shall be fitted. The ARMA models outperform

conventional time series models like moving average and other smoothing models in terms

of degree of forecast accuracy, treating seasonality and long term forecasts.

47

The disadvantage of the ARMA model is that it is tedious to build manually without the aid

of Statistical software.. Moreover, there are situations where the final model may not fit the

requirements due to error terms with non-constant variance. This is known as

heteroskedasticity, and may be treated separately using ARCH and GARCH techniques.

Another disadvantage of the ARMA compared to other conventional models is that forecasts

are unreliable with data series having less than 50 data points. The ARMA models are more

sensitive to outliers present in the original series. However, if the degree of accuracy is not

of great concern, conventional time series models can be employed that is simple and less

time consuming.

The proposed hybrid model (WT-ARMA) works as follows. The WT decomposes highly

nonlinear wind speed time series into several approximate stationary time series. The

decomposed time series are fed to different ARMA models established respectively, and

then the outputs of each ARMA model are combined to get the forecasting results. This is

shown in Figure 4.1. In order to see the effectiveness of the proposed approach, actual wind

speed data from a weather station are used to establish forecasting model. The results

indicate that the wavelet theory is a useful tool in wind speed forecasting and when

combined with the ARMA model brings significant improvement to reduce forecasting

errors.

4.2 Wavelet Analysis

Wavelet means a small wave or a pulse of short duration with finite energy that integrates to

zero. The Wavelet Transform divides the data into components of different frequency and

then matches resolution of each component to its scale. The WT breaks the original signal

into projections of translated and scaled versions of the original mother wavelet. The basis

function of the Wavelet Analysis is the mother wavelet function (ψ) [107-108].

48

Fig. 4.1 Procedure of Wind Speed Forecasting using WT-ARMA

The roots of the Wavelet Analysis have been derived from the very famous Fourier analysis

that mathematically transforms a stationary signal from time domain to frequency domain.

Although having a lot of advantages, the Fourier analysis has a serious drawback. When

transforming to the frequency domain, time information is lost and when transforming to the

time domain, frequency information is lost. That means both frequency domain and time

domain do not support and counter confirm each other’s data.

Reference [109] uses the WT to avoid this problem and improve the results for Fourier

transform. In this only a small section of the signal at a time is analyzed. This is called

windowing. The technique that maps a signal into a two-dimensional function of time and

frequency is known as Short-Time Fourier Transform (STFT). It balances the time and

frequency based views of a signal. So, it gives information about what time and frequencies

a signal event occurs but with limited precision.

Another limitation of STFT is that once a particular size for the time window is selected it

remains the same for all frequencies. Many signals demand for a more flexible approach

49

where changing the size of the window gives more accurate particular characteristics. The

wavelet transform gives better results and overcomes this deficiency, as it gives the option of

variable sized windows. Usage of long time intervals is allowed in the wavelet analysis.

Unlike Fourier Transform, the Wavelet transforms do not have a single set of basic

functions. The Fourier transform breaks the original signal into sine and cosine functions

waves of several frequencies whereas the wavelet transform breaks the signal into

projections of translated and scaled versions of the original mother wavelet. Thus, the

wavelet analysis provides immediate access to information that can be obscured by other

time-frequency methods. It generally has two categories as described in the following sub-

sections.

4.2.1 Continuous Wavelet Transform (CWT)

The continuous time wavelet transform of f (t) with respect to a wavelet (t) is given by

𝑊 (𝑚, 𝑛) = ∫ ∞

−∞𝑓 (𝑡)

1

√|𝑚| Ѱ ∗ [

𝑡 −𝑚

𝑛] 𝑑𝑡 (4.1)

where, m is the scale variable, n is the translation variable and * denotes complex conjugate.

The CWT maps the one dimensional function f (t) to a function W (m, n) having continuous

real variables m and n. The coefficient of W(m,n) at a particular scale and translation, tells us

how well the original function or signal f (t) matches with the scaled or translated mother

wavelet. However, to recover the function we do not require all the coefficients of W(m,n) so

CWT gives a redundant way to represent the signal [107,109].

The translated term relates to the window location that shifts through the signal. In the

transform domain it corresponds to time information. High scales relate to a non-detailed

global view (low frequencies) and low scale corresponds to detailed view (high frequencies).

4.2.2 Discrete Wavelet Transform (DWT) and Multi Resolution Analysis

(MRA)

DWT is used to decompose a signal into different resolution levels. Compared with CWT,

DWT is sufficient in decomposing and reconstructing most wind speed disturbances. It

50

provides enough information and offers high reduction in the computational time. Multi-

resolution analysis is to break a continuous real valued finite energy function into a

hierarchy of approximations. It is a technique that represents a function on many different

scales, which are formed by scaled and translated mother wavelet.

In order to obtain a matrix W of wavelet coefficients for Discrete Wavelet Transform

(DWT), it is possible to define w, in order to obtain a matrix W of wavelet coefficients,

which results in the Discrete Wavelet Transform (DWT)

𝑊 = 𝑤. 𝑥 (4.2)

This matrix w in equation (4.2) can be represented as 𝑤 = [𝑤1, 𝑤2, 𝑤2, … 𝑤𝐽, 𝑣𝐽 ]𝑇 .

Similarly, we can define W such as 𝑊 = [𝑊1,𝑊2, . .𝑊𝐽, 𝑉𝐽]𝑇

.Here, x being a time series

defined as 𝑥 = [𝑥1, 𝑥2, … 𝑥𝑁 ]𝑇with N an integer multiple of 2 𝑗 where j is the level of

resolution. For the DWT the first J sub- vectors contain all the wavelet coefficients for scale

J. Each 𝑤𝑗 column vector has 𝑁 2𝜏𝑗⁄ coefficients associated with changes on a scale of

length 𝜏𝑗 = 2𝑗−1 , for 𝑗 = 1, 2, 3, … , 𝐽 . The final sub-vector 𝜈𝐽 contains just the scaling

coefficients associated with averages on a scale of length 2 𝐽.

The DWT decomposes the signal into an approximation (low frequency) and detail (high

frequency) information and then analyzes the signal at different frequency bands with

different resolutions. DWT engages two sets of functions, called scaling functions and

wavelet functions, which are related with low pass and high-pass filters, respectively. By

successive high-pass and low-pass filtering of the time domain signal the breakdown of the

signal into different frequency bands is attained.

Multi Resolution Analysis (MRA) equation resulting from the reconstruction of the wavelet

coefficients is

𝑥 = 𝑤𝑇 .𝑊 = ∑ (𝑤𝑗𝑇𝐽

𝑗=1 .𝑊𝑗 ) + 𝑣𝐽𝑇 . 𝑉𝐽 = ∑ (𝐷𝑗) + 𝐴𝐽

𝐽𝑗=1 (4.3)

In the MRA equation (4.3), the time series x is stated as the sum of a constant vector AJ and J

other vectors, Dj, (j=1,2,3...J), each of which contains a time series related to variations in x

at a certain scale. The Dj refer to the jth wavelet detail and the AJ as the approximation. MRA

51

is intended to give good time resolution and poor frequency resolution at high frequencies

and good frequency resolution and poor time resolution at low frequencies. This

methodology has proved to give good results especially when the signal has high frequency

components for short durations and low frequency components for long durations.

1. The decomposition of algorithm: According to the decomposition algorithm, if A0 is a

discrete signal to be decomposed, then the equation is

{𝑎𝑗+1 = 𝐻𝑎𝑗

𝑑𝑗+1 = 𝐺𝑑𝑗 (4.4)

where, j= 0 -J, J is the maximum decomposed level; H is a low-pass filter; G is a high

pass filter; a j and dj are respectively low frequency signal and high frequency signal of

the original signal in the resolution of 2-j with the adjacent of the original signal in the

different frequency band. Finally, a0 is decomposed into d1,d2...dJ and aJ. To make the

total output data length and the length of the input data remain consistent the

decomposition algorithm makes the data length half in each layer than the one before the

decomposed signal [110].

2. The reconstruct algorithm: The expression of the reconstruct algorithm is

𝐴𝐽 = 𝐻∗𝑎𝑗+1 + 𝐺∗𝑑𝑗+1 (4.5)

where, j =J-1, J-2...0 ; H* and G* are the dual operators. The decomposed wavelet

increases the number of signals which are later reconstructed using the Equation 4.5.

Reconstruct d1, d2,..., dJ and aJ separately to get D1,D2,...,DJ and AJ ,then X can be yielded

𝑋 = 𝐷1 + 𝐷2 …𝐷𝐽 + 𝐴𝐽 (4.6)

where D1 ={d1,1,d1,2,...,}..., DJ= {dJ,1,dJ,2,..} is the reconstruction of high frequency

signals from the first layer to the Jth layer; AJ ={aJ,1,aJ,2,...} is the reconstruction of low

frequency signals of J layer.

52

4.3 AUTO REGRESSIVE MOVING AVERAGE (ARMA)

Wind pattern expresses an equivalent theory in wind forecasting. A wind pattern illustrates

one specific type of trend of wind process at a particular site or location, and duration of

each pattern may be special for that particular region. When the traits of wind patterns are

known, forecasts of wind speed become easier in the located area. The wind patterns can be

used as complements to forecasts based on the ARMA, which are blind to the possible trend

of wind speed variation in the following hours.

The ARMA models can be described by a series of equations. The mean-adjusted series are

obtained if the time series is first reduced to zero-mean by subtracting the sample mean.

Therefore, we will work with the mean-adjusted series

𝑦𝑡 = 𝑌𝑡 − ��, 𝑡 = 1,…𝑁 (4.7)

where 𝑌𝑡 is the original time series, �� is its sample mean, and 𝑦𝑡 is the mean-adjusted series.

One subset of the models is the so-called autoregressive, or AR models. This model

describes a time series as a linear function of its past values. The order of the AR model

signifies the number of lagged past values included. The simplest AR model is the first-order

autoregressive, or AR (1), model

𝑦𝑡 + 𝜑1 𝑦𝑡−1 = 𝑎𝑡 (4.8)

where yt is the mean-adjusted series in year t, 𝑦𝑡−1 is the series in the previous year, 𝜑1 is

the lag-1 autoregressive coefficient, and at is the noise. The noise is also known as the error,

the random-shock, and the residual. The error at is supposed to be random in time, not auto

correlated, and normally distributed. We can rewrite the equation for the AR (1) model as

𝑦𝑡 = − 𝜑1 𝑦𝑡−1 + 𝑎𝑡 (4.9)

This equation shows that the AR (1) model has taken the form of a regression model in

which yt is regressed on its previous value. Here, a1 is comparable to the regression

coefficient, and at to the regression residuals. The name autoregressive implies to the

regression on self (auto). Higher-order autoregressive models include more lagged yt terms

as predictors.

53

The moving average (MA) model is a form of the ARMA model in which the time series is

regarded as a moving average (unevenly weighted) of a random shock series at. The first-

order moving average, or MA (1), model is given by

𝑦𝑡 = 𝑎𝑡 + 𝜃1𝑎𝑡−1 (4.10)

where at, at-1 are the residuals at times t and t-1, and 𝜃1 is the first-order moving average

coefficient. MA models of higher order than one include more lagged terms.

This shows that the AR model includes lagged terms on the time series itself, and that the

MA model includes lagged terms on the noise or residuals. So, by taking both types of

lagged terms, we arrive at what is called autoregressive-moving-average, or ARMA, model.

The order of the ARMA model is included in parentheses as ARMA (p, q), where p is the

autoregressive order and q the moving-average order. The simplest ARMA model is first-

order autoregressive and first-order moving average, or ARMA (1,1). The ARMA model is

used for stationary time series that take past values, prediction errors and a random term into

account.

An ARMA (p, q) model of order p and q, with 𝜑𝑗and 𝜃𝑗 the autocorrelation and the moving

average coefficients respectively, is represented by [108]

𝑦𝑡 = ∑ 𝜑𝑗 𝑝𝑗=1 𝑦𝑡−𝑗 + 𝑎𝑡 − ∑ 𝜃𝑗

𝑞𝑗=1 𝑎𝑡−𝑗 (4.11)

The process of the ARMA(p, q) is combined by the processes of AR (p) and MA (q). If q=0,

then the equation becomes an AR model of order p. When p=0, the process becomes an MA

model of order q. In the ARMA (p, q) model, {yt} is the return series of original time series

and {at} is the innovations noise process. In the process of constructing an ARMA model, it

is imperative to confirm the orders p and q.

Equation 4.11 is put into Equation 4.3 to get the proposed hybrid formula that is

𝑥 = ∑ 𝐷𝑗 + 𝐴𝐽𝐽𝑗=1 = ∑ 𝐷𝑗

𝐽𝑗=1 + {𝑦𝑡 − ∑ 𝜑𝑗

𝑝𝑗=1 𝑦𝑡−𝑗 + ∑ 𝜃𝑗

𝑞𝑗=1 𝑎𝑡−𝑗} (4.12)

This explains the proposed hybrid. The original wind speed data are taken and put into the

Wavelet Transform that first decomposes the highly non-linear data and then reconstructs it.

54

It is then fed into ARMA, which gives the combined forecasting data with very less errors.

In this study, an index, namely mean absolute percent error (MAPE), is used as forecasting

precision measure. The index is represented as follows:

𝐸𝑀𝐴𝑃𝐸 =1

𝑁 ∑ |

𝑌𝑡−𝑌��

𝑌𝑡|𝑁

𝑡=1 (4.13)

where Yt is the actual time series value at period t, 𝑌�� is the forecasting time series value at

period t, and N is the number of forecasting periods.

4.4 Results and Discussion

In the present study, actual wind data and the input-output patterns for training of a network

are taken from Auckland City, New Zealand, for the year 2011 on a 30 seconds interval. This

dataset was provided by NIWA, which is the only producer of wind speed datasets in New

Zealand. Only wind speed and direction were provided therefore no other factors such as

temperature are considered here. The platform chosen to test the available set of data is

MATLAB. The forecasting errors are also computed using MATLAB.

A hybrid model is established on the basis of consecutive hourly mean wind speed samples,

thus allowing the prediction of wind speeds at future time-steps.

Firstly, the original wind speed time series is decomposed into detailed time series at

different resolutions (scales), d1, d2,..., dJ and a coarse approximation time series aJ, then we

obtain Dj (1 ≤ j ≤ J) and AJ by reconstruction. From this, the data is being filtered. If direct

data are put into the ARMA model then the results will not be very accurate [108]. As a

result, for different time series (Dj and AJ) of Wavelet Transform, the ARMA models are

established to forecast respectively. Lastly, the hybrid approach is utilized to obtain the

forecasting result. In the experiments, 250 input/output data values are used for the

simulation. The first 215 data values form the training data set, whereas the others are used

as checking data for validating the wavelet-ARMA model.

Figure 4.2 is plotted for WT, which is the sum of the decomposition at level 5 and

reconstruction of the wavelets. The approximations for the low frequency and details for

55

high frequency are added and the summation of both is shown in this figure. It shows the

decomposed coefficients of every scale. D1, D2,.. D5 are details, they reflect the detail

information of every scale. A5 is the approximation; it reflects the development trend of wind

speed. The relation between A5 and Dj (1≤ j≤ 5) meets S=A5+D5+D4+D3+D2+D1.

The characteristic of wavelet decomposition is that the more scales the original signal can be

decomposed into, the better signals are, but great errors will be brought about at the same

time. As a compromise the decomposed number of scales is always less than 6 [107-109].

Therefore, data were tested for j = 1-6 levels for the hybrid and j = 5 was seen to be giving

the best results therefore j = 5 has been chosen. Table 4.1 shows the different errors (MAPE)

obtained for different values of j and hence results show why j=5 has been chosen for testing

this particular dataset in this section.

Table 4.1 Different Error for Different values of 1 ≤ j ≤ 6

j WT-ARMA

MAPE(%)

1 3.4

2 3.2

3 3.0

4 3.0

5 2.7

6 2.9

Now the ARMA model is fed with the wavelet transform output and the results analysed for

different p and q. The results of ARMA(1,1), ARMA(1,2), ARMA(2,1), and ARMA(2,2) are

compared. However, no significant discrepancies between the results are observed. As an

example, the results for ARMA(1,1) are displayed in Fig. 4.3. One of the possible reasons

for not observing any difference could be due to the use of synthetic data.

56

Fig. 4.2 Output of Wavelet Analysis shows approximations for

the low frequency (A5) and details for high frequency Dj (1≤ j≤ 5)

that are added as (S).

After the parameters are learned from the training set, the forecasting algorithm is applied as

described in Section 3 to make 35 points’ predictions (1-hour-ahead prediction). Figure 4.4

compares the forecasted results of WT-ARMA hybrid to the original wind. Figure 4.5

shows the Wavelet ARMA hybrid with the original wind data and gives a very good

minimum error.

57

Fig. 4.3 Output of ARMA fed with Wavelet

These results have been compared with the Artificial Neural Networks (ANN) - the

Ensemble Kalman Filter (EnKF) hybrid [111]. The MAPE of the ARMA has been shown to

be between 7% for 1-hour [11] as shown in Table 4.2. It shows the MAPE and the

computational time for ARMA, ANN-EnKF and Wavelet-ARMA techniques. It clearly

shows the MAPE for the Wavelet- ARMA hybrid is the smallest and thus proving it to be an

output correction scheme.

58

Fig. 4.4 Actual Wind Speed versus Forecasted using Hybrid

Model

Fig. 4.5 Error graph for WT-ARMA hybrid

59

Table 4.2 Results Analysis

Comparison between Wavelet ARMA hybrid and ANN-KF in Wind Speed Forecasting

Method ARMA ANN-KF Wavelet –ARMA

MAPE% 7.98% 6.57% 2.7%

Computational Time 100 sec 180 sec 160 sec

Table 4.2 shows the MAPE and Computational Time of ARMA, ANN-KF and WT-ARMA.

The MAPE is the least of WT-ARMA since this hybrid proves to be a better one in this

research. The computational time of ARMA is the least because of less complexity as

compared to the hybrids as in [128]. It can be seen that the MAPE of the Wavelet-ARMA

model is 2.70%, marginally outperforming the persistence model, ARMA, ANN-EnKF

model.

4.5 Summary

This chapter presents a very effective WT-ARMA hybrid forecasting technique. The

nonlinear wind speed time series is seen to be decomposed which is later fed to different

ARMA models which are then reconstructed to get new and improved results. A unique

equation that combines both these methods is established. This hybrid has also been tested

on MATLAB. In order to see the effectiveness of the proposed approach, different figures

for individual as well as the proposed hybrid technique has been plotted. The results indicate

that the wavelet theory when combined with the ARMA model becomes a beneficial tool in

wind speed forecasting and brings a tremendous amount of improvement to reduce

forecasting errors.

60

Chapter 5

ARMA-ANN-AIC Hybrid

5.1 Introduction

This Chapter focuses on developing a novel wind speed forecasting technique, which

produces more accurate predictions. A hybrid method composed of the Artificial Neural

Network (ANN) and Akaike Information Criteria (AIC) along with the Auto Regressive

Moving Average (ARMA) technique is proposed. One of the main challenges is to properly

determine the order of ARMA that would optimize the results. This research combines

ANN with AIC to determine the true order of ARMA and bring down the forecasting error.

Simulation studies have been conducted to show the effectiveness of the proposed method

and have been compared to GA. The results have shown very good results in the forecasting

error for the test data given in this part.

5.2 ARMA

An ARMA (p, q) model of order p and q, with 𝜑𝑗and 𝜃𝑗 the autocorrelation and the moving

average coefficients respectively, is represented by [108]

𝑦𝑡 = ∑ 𝜑𝑗 𝑝𝑗=1 𝑦𝑡−𝑗 + 𝑎𝑡 − ∑ 𝜃𝑗

𝑞𝑗=1 𝑎𝑡−𝑗 (5.1)

61

The order of the ARMA model is included in parentheses as ARMA (p, q), where p is the

autoregressive order and q the moving-average order. In Equation (5.1), {y (t)} is the return

series of original time series. In the process of constructing an ARMA model, it is imperative

to confirm the orders p and q [108].

ARMA parameters may be obtained from three-layer neural networks utilizing a polynomial

representation of the activation function in the hidden units. Consider a nonlinear, time-

invariant, discrete-time dynamic system represented by the following ARMA model shown

in Fig. 5.1.

5.3 ANN

Figure 5.1 shows a three-layer ANN topology. Note that weights of x input neurons are W

leads and weights of y input neurons are V leads. The coefficients are obtained from the

neural network weights values and polynomial coefficients given by

𝐚𝐢 = ∑ 𝐖𝐢𝐣𝐌𝐣=𝟏 𝐚𝐣 𝐕𝐢𝐣 𝐲(𝐧 − 𝐢) (5.2)

𝐛𝐢 = ∑ 𝐖𝐢𝐣𝐌𝐣=𝟏 𝐚𝐣 𝐕𝐢𝐣 𝐱(𝐧 − 𝐢) (5.3)

where M is the number of hidden units, i, and n are indexes [112].

62

Fig. 5.1 A three layer ANN topology. Note that the weights of x input

neurons are W leads and the weights of y input neurons are V leads.

These model parameters are obtained using ANN, which are then fed into the AIC to obtain

the true order.

5.4 AIC

The AIC criterion can be defined as

𝐀𝐈𝐂(𝐝) = 𝐥𝐨𝐠〖(��𝐝𝟐〗) + 𝟐

𝐝

𝐍 (5.4)

where 𝜎𝑑2 is the error variance of fitted model and N is the number of observations estimated

under the assumption that d is the true order [113-114].

63

For the ARMA (p, q) model, d is the model order where d = p + q. In practice, the optimal d

is obtained by minimizing AIC (d) where

σd2 =

1

N∑ e(n)2N

n=1 (5.5)

where,

𝐞(𝐧) = 𝐲(𝐧) − ��(𝐧) (5.6)

and ��(𝐧) is given by

y(n) = −∑ aky(n − k)tk=1 (5.7)

The value of AIC is obtained for different p and q. The one with the least value of AIC is

considered to be the true order. It can be explained with the help of Fig. 5.2.

According to Fig. 5.2, the parameters are obtained using Equation (5.2) and (5.3) of ANN.

Then these model parameters are fed into Equation (5.4) of Akaike to estimate the true order

of ARMA. Then, ARMA takes the true order and forecasts wind speed data. For the sake of

completion, it is imperative to account for the following steps

1) First, we fix the range of the ARMA model orders to be considered (1 ≤ p ≤ pmax and 1

≤ q ≤ qmax)

2) Then, estimation of the model parameters by an appropriate algorithm such as ANN is

done. The predicted signal ��(𝒏) is computed keeping the error variance σd2 equal to one

mimicking what has been done in [112].

3) Evaluate AIC over S=(1 ≤ p ≤ pmax), and choose the value of d that minimizes AIC to

find the true order of ARMA.

64

Fig. 5.2 Procedure of the ANN-ARMA-AIC hybrid

5.5 Results and Discussion

In this research, wind data and the input-output patterns for training of a network are taken

from North Island, New Zealand for the year 2006 on a hourly basis at a 30 seconds interval

for station number STH1, as provided by NIWA 1. The platform chosen to test the available

set of data is MATLAB. The six hourly wind fields are extracted from the data. The test data

have an error of 0.8%.

During the experiments, 1500 input/output data values are employed for the simulation. The

first 750 data values are used for the training (these become the training data set), while the

others are used as checking data for validating the ARMA model.

1 NIWA is a Crown Research Institute of NZ.

65

For the first part ANN is used to determine the model parameters. We use a polynomial

function neural network (PFNN) model for training the simulated data to identify the

coefficients of ARMA models. The max values of p and q have been taken equal to 6, giving

a total of 36 possible combinations for p and q. The best order has been found to be (p=3,

q=2) as shown in Table 5.1 therefore the results for this order have been shown here. For

this order the ARMA model was generated with the Moving Average (MA) excitation being

uncorrelated Gaussian white noise (GWN) with a variance of one as in [130]. The following

linear ARMA model was utilized:

y(n) = x(n) + 0.358 x(n−1) + 0.212 x(n−2) + 0.343 x(n−3) + 0.201 y(n−1) +

0.301 y(n−2) (5.8)

For the PFNN analysis, the input and output data pair was segmented into two 750-point

data segments. The first half of the input–output data segment has been used to train the

network, and the second half was used to test the predictive quality of the network. All of the

simulations were carried out in this manner. Since the PFNN method utilizes a back-

propagation learning algorithm, the time required to compute parameter estimates is less

than GA. It should also be noted that the learning rate of the neural network is affected by

the selection of the ARMA model order (or the selection of the memory length), that is,

longer training of the network is required with larger ARMA model orders, and vice versa.

To examine how strongly the neural network topology depends on the assumed model order,

the results obtained by the PFNN method and the Genetic Algorithm have been compared.

To check the validity of the results from Equation 5.2 and 5.3, the simulations were also

carried out for different combinations of 1 ≤ p ≤ 6 and 1 ≤ q ≤ 6. The results for the order

p=3 and q=2 were chosen, corresponding to the best case as it will be shown in the latter

section. Table 5.2 shows that both methods gave very similar parameter values.

Using Table 5.2, the time series were computed corresponding to the parameter values

obtained for ANN and GA. The difference between the two time series is plotted in Fig. 5.3.

This figure shows that there is no significant difference between the two outcomes thereby

indicating the validity of the proposed approach when compared to GA.

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TABLE 5.1 THE EXHAUSTIVE SEARCH RESULTS FOR DIFFERENT

p AND q

q

p

1 2 3 4 5 6

1 329.3 329.4 328.9 330.0 330.3 330.4

2 328.1 329.7 329.8 328.5 330.3 331.2

3 327.1 325.9 326.5 327.3 328.9 330.3

4 333.0 331.2 333.3 334.1 332.4 337.1

5 335.2 335.4 337.5 339.1 340.1 340.5

6 340.3 342.4 341.3 344.4 343.5 346.6

TABLE 5.2 COMPARISON OF PARAMETERS USING ANN AND GA

Parameters Parameters for order (3,2)

a1 a2 a3 b1 b2

ANN 0.3 0.21 0.34 0.2 0.3

GA 0.35 0.21 0.3 0.21 0.3

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Fig. 5.3 Difference between the parameters for ANN and GA

Next, these parameter values are put in the equation of AIC to determine the true order of

ARMA. An exhaustive search for different p and q has also been done to compare the

results. It is shown in Table 5.1. The matrix gives the AIC values for different p and q

values. It can be seen that the least values are obtained for p = 3 and 1 ≤ q ≤ 4. The least

value is obtained for p = 3, q = 2. The other combinations have a greater AIC value.

The procedure for true order determination by AIC is summarized as:

1. The different parameters are fed into AIC.

2. The different values of AIC are determined for different p and q orders.

3. The order that gives the least value of AIC is considered to be the best order.

68

The AIC values have been calculated for 1 ≤ p ≤ 6 and 1 ≤ q ≤ 6. The range of AIC values

obtained is between 326 and 345. The best values are observed to be between p = 3 and 1 ≤

q ≤ 4 and these AIC values are given in Table 5.3.

TABLE 5.3 AIC VALUES

Model Order AIC for p=3, 1 ≤ q ≤ 4

AIC

(3,1) 327.1

(3,2) 325.9

(3,3) 326.5

(3,4) 327.3

When ARMA is applied to the test data of station STH1 provided by NIWA with p = 3 and

q = 2, the forecasting error is close to 0.75% which is very close to actual error provided by

NIWA. If we propose a hybrid including ARMA with the correct p and q and another

method then it may bring down the error even further, which opens doors for future

researches.

5.6 Summary

This Chapter presents a parameter estimation problem, which is necessary for ARMA to run

effectively. In order to do so, this research combines ANN-ARMA-AIC to estimate

parameters properly. First, the order is estimated with the help of ANN which is later fed

into AIC which estimates the true order of ARMA. The order that gives the least value of

AIC is considered to be the best order. This true order is given as an input into ARMA,

which makes ARMA to function effectively. The simulation studies were also carried out for

different combinations of 1 ≤ p ≤ 6 and 1 ≤ q ≤ 6. The results for the orders p = 3 and q = 2

were chosen, corresponding to the best case. The hybrid is tested on data taken from North

Island, New Zealand for the year 2006 on MATLAB. The results show a very low

forecasting error, which proves the proposed hybrid, is a successful one.

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Chapter 6

ORPD and PSO in GT

6.1 Introduction

The objective of this Chapter is to find the correlation between the forecasting and its

application in power systems. Therefore, the Economic Dispatch (ED) is identified as

another problem with wind power. Wind power may be dispatched above or below its

required demand during periods of excessive or insufficient supply. When it is dispatched

above its required demand, the Energy Storage System (ESS) comes into play. It stores the

extra wind power, which can be used in times of insufficient supply. The ESS is used to

compensate the power fluctuations from the wind farm. The ESS is commanded to supply or

absorb power equal to the fluctuations between the original output of the wind power

generator and the desired output of the ESS conjunction with the wind power generator in a

particular regulation period.

It is usually quite difficult to determine how to dispatch the power, which should result in

minimum losses. Wind energy and its forecasting may act as the most effective solution to

the ED problem therefore it becomes crucial that the wind forecast is accurate. The ED

problem is due to the voltage instability because when voltage fluctuates the economic

dispatch is not proper [129]. Therefore, Optimal Reactive Power Dispatch (ORPD) should

be used in order to reduce active power and voltage losses. This research mainly focuses on

the ORPD and GT. The voltage instability is not done here it being a completely different

area of research.

70

6.2 Optimal Reactive Power Dispatch

ORPD plays a significant role in optimal operation of electric power systems. It is a complex

nonlinear optimization problem with a mixture of continuous and discrete control variables.

ORPD is a sub- class of the optimal power flow (OPF) problem [92, 115]. The objective of

ORPD is to minimize the transmission losses and to control the voltage profile such that the

voltage deviations at the load buses for various loading conditions. ORPD plays a vital role

for the secured economic performance of power systems.

This chapter describes an approach to the ORPD problem. PSO is one of the evolutionary

computation technique based on swarm intelligence. It is sensitive to the tuning of its

parameters and has a flexible mechanism to explore a global optimum point within a short

calculation time [115]. DE and PSO are population-based optimization algorithms. Due to

their excellent convergence characteristics and few control parameters, DE and PSO have

been applied to obtain optimal solutions to some real value problems efficiently [116].

The following flowchart (Fig. 6.1) shows the sequence of problems that are addressed in the

study. If wind energy is unable to be predicted efficiently then the ED problem arises due to

voltage instability [25]. Then, ORPD comes into play for which PSO is used as a solution in

this research. The input data are taken from a previous Auto Regressive Moving Average

(ARMA) and Wavelet Transform (WT) hybrid [Figure 4.4] [117] and is tested on IEEE-14

Bus system in GT.

The objective of the reactive power dispatch is to minimize the active power loss in the

transmission network, which can be described as follows [22, 80, 115]:

𝑚𝑖𝑛 ∑ 𝑃𝑘𝑙𝑜𝑠𝑠 = ∑ 𝑔𝑘(𝑣𝑖2 + 𝑣𝑗

2 − 2𝑣𝑖𝑣𝑗 cos 𝜃𝑖𝑗)𝑘=𝑁𝐸𝑘=𝑁𝐸 (6.1)

where k= (i, ,j); (Total number of buses), (No. of buses adjustment to bus i

including bus i, NE= Set of numbers of network branches, = Total active power losses

iÎNB j ÎNi

Pklossk=NE

å

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in the transmission system, gk = conductance of branch k (p.u.), vi, vj = voltage magnitude

(p.u.) of bus i and j resp., θij = load angle difference between bus i and j (rad).

Fig. 6.1 Flowchart showing the sequence of problems in ED

Equality constraints:

Active power flow balance equations at all buses

𝑃𝑔𝑖 − 𝑃𝑑𝑖 − 𝑣𝑖 ∑ 𝑣𝑗(𝑔𝑖𝑗 cos 𝜃𝑖𝑗 + 𝐵𝑖𝑗 sin 𝜃𝑖𝑗) = 0𝑗=𝑁𝑖 (6.2)

where Pgi is the injected active power at bus i (p.u.), Pdi is the Demanded active power at bus

i (p.u.), vi is the voltage magnitude of bus i (p.u.), gij is the Transfer conductance between

bus i and j (p.u.), Bij is the Transfer susceptance between bus i and j (p.u.)

Reactive power flow balance equations at all PQ buses (load buses)

72

𝑄𝑔𝑖 − 𝑄𝑑𝑖 − 𝑣𝑖 ∑ 𝑣𝑗(𝑔𝑖𝑗 sin 𝜃𝑖𝑗 + 𝐵𝑖𝑗 cos 𝜃𝑖𝑗) = 0𝑗=𝑁𝑖 (6.3)

where Qgi is the Injected reactive power at bus i (p.u.), Qdi is the Demanded reactive power at

bus i (p.u.).

Inequality constraints:

Reactive power generation limit for each generator bus

𝑄𝑔𝑖𝑚𝑖𝑛 ≤ 𝑄𝑔𝑖 ≤ 𝑄𝑔𝑖

𝑚𝑎𝑥 , 𝑖 ∈ 𝑁𝑔 (6.4)

Voltage magnitude limit for each bus

𝑣𝑖𝑚𝑖𝑛 ≤ 𝑣𝑖 ≤ 𝑣𝑖

𝑚𝑎𝑥 , 𝑖 ∈ 𝑁𝑔 (6.5)

Power flow limit constraint of each transmission line

𝑠𝑙 ≤ 𝑠𝑙𝑚𝑎𝑥 (6.6)

where Sl is the Power flow in branch l (p.u.).

6.3 Particle Swarm Optimization (PSO)

The Particle swarm optimization (PSO) method was introduced by Kennedy and Eberhart

[24]. It was described as a self-educating stochastic optimization algorithm that could be

applied to any linear, nonlinear, mixed-integer optimization problems having continuous

and/or discontinuous objective and constraint functions. PSO has many obvious advantages

over the other evolutionary algorithms. It has a high probability of finding a global

minimum. It is a fast, simple and efficient population-based optimization method. PSO does

not require parameter tuning and gives a faster and tougher optimal search [16].

It is a multi-agent search technique that traces its evolution to the emergent motion of a flock

of birds searching for food [115, 118]. PSO finds the optimal solution by a process

73

motivated by the imitation of social behavior such as fish schooling and birds flocking in

search of food and not by survival of the fittest. A population of random potential solutions

called particles that come together to form a group known as a swarm initializes PSO. Each

particle in the swarm has a certain velocity that helps it move over the search space. The

velocity of each particle is influenced by the particles’ own flying experience as well as its

neighbors’ flying experience. Every agent’s position is represented along XY axis and Vx

and Vy respectively represent the velocity. Each agent knows its local best value so far (pbest)

and global best value in the group (gbest). Namely, each agent tries to modify its position

using the current positions (x, y); the current velocities (Vx, Vy); the distance between the

current position and pbest; the distance between the current position and gbest [26]. PSO

solves a highly constrained problem as an unconstrained one by adding a suitable penalty

function to the main objective function [16].

In computer science a set of randomly generated solutions propagates in the design space

towards the optimal solution over a number of iterations. PSO optimizes a problem by

having a population of candidate solutions, here dubbed particles, and moving these particles

around in the search-space according to simple mathematical formulae over the particle's

position and velocity. Each particle's movement is influenced by its local best-known

position and is also guided toward the best-known positions in the search-space, which are

updated as better positions are found by other particles. This is expected to move the swarm

toward the best solution as shown in the following equations [17- 18].

𝑣𝑖𝑘+1 = 𝑤 × 𝑣𝑖

𝑘 + 𝑐1 × 𝑟1 × (𝑝𝑏𝑒𝑠𝑡𝑖− 𝑥𝑖

𝑘) + 𝑐2 × 𝑟2 × (𝑔𝑏𝑒𝑠𝑡 − 𝑥𝑖𝑘) (6.7)

𝑥𝑖𝑘+1 = 𝑥𝑖

𝑘 + 𝜒 × 𝑣𝑖𝑘+1 (6.8)

where, vik+1

is the velocity of ith particle at (k +1)th

iteration; w is the inertia weight of the

particle; vik

is the velocity of ith particle at kth iteration; c1 and c2 are the acceleration

constants having values between [0, 2.5] on an average give better performance, and also,

helps find the global optimum within a reasonable number of iterations; r1 and r2 are

randomly generated numbers between [0, 1]; pbest is the best position of the ith particle

obtained based upon its own experience; gbest is the global best position of the particle in the

population; xik+1

is the position of ith particle at (k +1)th

iteration; xik is the position of ith

74

particle at kth iteration; c is the Constriction factor, which is a function of c1 and c2 as

shown below. It may help insure convergence [17].

𝜒 =2

|2−𝑐−√𝑐2−4𝑐| (6.9)

where c = c1 + c2 and c > 4

In general, inertia weight factor w decreases linearly from max w1 to min w2 according to the

following equation

𝑤 = 𝑤𝑚𝑎𝑥 −𝑤𝑚𝑎𝑥−𝑤𝑚𝑖𝑛

𝑖𝑡𝑒𝑟𝑚𝑎𝑥× 𝑖𝑡𝑒𝑟

(6.10)

Where, wmax is the value of inertia weight at the beginning of iterations, wmin is the value of

inertia weight at the end of iterations, iter is the current iteration number and itermax is the

maximum number of iterations.

6.4 Wavelet Transform and ARMA Hybrid

The input data for this PSO has been taken from Chapter 4, which uses the hybrid of WT and

ARMA for forecasting wind speed [117]. This research is in continuation of this hybrid to

see the impact of accurate forecasting on the economics. The hybrid equation used (eq 4.12)

in Chapter 4 is:

𝑥 = ∑𝐷𝑗 + 𝐴𝐽

𝐽

𝑗=1

= ∑ 𝐷𝑗

𝐽

𝑗=1+ {𝑦𝑡 − ∑𝜑𝑗

𝑝

𝑗=1

𝑦𝑡−𝑗 + ∑𝜃𝑗

𝑞

𝑗=1

𝑎𝑡−𝑗}

where the time series x is stated as the sum of a constant vector AJ and J other vectors, Dj

(j=1,2,3...J), each of which contains a time series related to variations in x at a certain scale.

The Dj refers to the jth wavelet detail and the AJ as the approximation. In the ARMA (p, q)

model, {yt} is the return series of original time series and {at} is the innovations noise

75

process. The order of the ARMA model is included in parentheses as ARMA (p, q), where p

is the autoregressive order and q the moving-average order. Also, φj and θj are the

autocorrelation and the moving average coefficients respectively.

6.5 Graph Theory (GT)

A GT has an excellent capability to simplify large power networks. The GT can solve a

number of things such as shortest path algorithm in a network, finding a minimum spanning

tree, finding graph planarity, algorithms to find adjacency matrices, algorithms to find the

connectedness just to name a few. It also helps to detect and isolate faults. When a node fault

is detected, the corresponding row is eliminated corresponding to the fault-node(s) and the

columns that correspond to the edges that connect these nodes. The edges surrounding the

fault-node(s) are detected and are placed into a row matrix in order to be returned by the

function and later disconnected. One can also shed unbalanced loads. If the generators

capacity is not enough to supply all of loads the algorithm will disconnect the loads with

lower priorities [25].

A number of power systems properties can be appropriately defined by means of a diagram

consisting of a set of vertices together with a set of edges that may be joined together.

A graph may be directed or undirected. An undirected graph is one in which there is no

distinction between the two vertices associated with each edge. A directed graph shows a

flow network from node to node where each edge acts as an input or output for the flow of

current, voltage etc. The amount of flow on an edge cannot exceed the capacity of the edge.

The amount of flow entering into a node should equal the amount of flow out of that node

[119]. This type of graph may be used to manage fluid in pipe or currents in an electrical

circuit for instance [120]. It is also known as a network, the vertices are called nodes and the

edges are called arcs. The following are few types of graphs-

1. Directed Graph: is represented as G=( N, E), consists of a set N of nodes (or vertex) and

a set E of directed arcs (or edges) whose elements are ordered pairs of distinct nodes

2. Node-Edge Incidence matrix: is represented as an nxm matrix, which contains one row

for each node of the directed graph and one column for each edge. The column

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corresponding to arc (i, j) has only two nonzero elements, -1 in the row

corresponding to node i and +1 in the row corresponding to node j

3. Network: A network is a graph or digraph with weights assigned to each edge. These

numbers might represent distance, capacity, resistance, etc., depending on the

context.

4. Net flow: In an arc (i, j), a net flow is a real-valued function f (i, j), which may be thought

of as an amount of some commodity that can arrive to j from i per unit time, this

could be positive or negative. A conservation criterion at each node must be satisfied

i.e., the total flow into a node must be equal to the total flow out of the node, unless

the node is a source, which has more outgoing flow, or sink, which has more

incoming flow.

5. Capacitated Network: Is a network in which some arcs are assigned nonnegative

capacities, which define the maximum allowable flow in those arcs. The capacity of

an arc (i, j) is denoted kij, and this capacity is indicated on the graph by placing the

number kij adjacent to the arc.

6. Residual Network: Given a flow network, the residual network consists of arcs that can

admit more net flow. The amount of available capacity of an arc (i, j) is the residual

capacity and is given by krij = kij – fij.

A directed graph has been used in this research. In the graph G, the elements vi ÎV , i =

1,2,...,nb, and eij ÎE ÌV ´V , i, j = 1,2,...,nb, denote the set of nodes and edges, respectively.

The sets V and E represent the buses and branches in the power system, respectively. This

may be used for the evaluation in a wind farm where there is a path joining every turbine in

the wind farm (which produces electricity) with the reference node, thus making it relatively

easy. If there is a possible path it means that the energy produced by the turbine can be

dispatched to the electrical system, otherwise that energy is lost [121-122].

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6.6 Results and Discussion

MATLAB has become a popular tool for scientific computing and is well suited for the

numerical computation typical of steady-state power system simulations. This thesis uses

MATPOWER, an open-source MATLAB power system simulation package [123-124]. It is

used widely in research and education for AC and DC power flow and OPF simulations.

MATPOWER consists of a set of MATLAB M-files designed to give the best performance

possible while keeping the code simple to understand and customize. It combines a high-

level language ideal for matrix and vector computations, a cross-platform runtime with

robust mathematical libraries, an integrated development environment and GUI with

excellent visualization capabilities, and an active community of users and developers [116].

This research has been tested on the IEEE 14 bus system, which is a standard grid system,

used for testing any power unit or fault of any power system components. Simulation of the

modified IEEE 14 bus system has been built as follows. At bus number 3, Doubly Fed

Induction Generator wind turbines are connected as shown in Fig. 6.2. Bus number 3 has

been chosen randomly since at any bus the results may be the same. Very less power is lost

via the converter, which makes DFIG very popular in improving system performance. In this

system, there are four Generators that act as a source. They are located at Bus 1, 2, 6 and 8

respectively. PV (Active Power and Voltage) bus at Bus 2 is the bus where voltage is being

controlled. There are three condensers at Bus 3, 6, 8. TG refers to the Turbine Governor at

Bus 2. It controls the flow rate of steam turbine so as to maintain its speed of rotation as

constant. AVR refers to the Automatic Voltage Regulators that help in regulating voltage.

Also, there are four transformers connected between Bus 4-9, 4-7, 5-6, and 7-8. The slack

bus, which is the reference bus, is taken to be Bus 1 in this research. The losses in Table 6.1

are calculated with respect to this slack bus. These combined together form the IEEE-14 Bus

System used in this study.

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Fig. 6.2 Standard IEEE- 14 bus system with DFIG at Bus 3

The corresponding graph is shown in Fig. 6.3 where each bus corresponds to a node. It

shows the relationship between nodes thus showing that the data may flow unidirectionally.

Also, since the network is designed to be redundant, it makes use of the unidirectional ability

79

to cope with failure. As node property, each node has input ports and output ports, so the

flow entering a loop should be equal to the flow exiting a loop.

Fig. 6.3 Graph showing the relationship of Power Flow

This graph is making three things very simple. Firstly, it provides easy communication,

where information in the form of electric signals is taking place in both directions. Secondly,

there can be a Reduction of Loss of Electricity since finding the shortest path can shorten the

distance of transfer of electricity. For instance, if node 13 is acting like the input node and 14

as the output node, then the shortest path may be from 13->9->2->14. This will help reduce

electricity loss. Thirdly, if any node or Bus is out of service then, that node can be isolated

80

and the fault may be avoided. Suppose, if node 7 stops working then isolate it and still

continue with 13->9->2->14 to find and work with the shortest path.

To calculate the losses, PSO has been used where the input data are taken from the results of

the ARMA and WT hybrid shown in Fig. 4.4 from Chapter 4 [117].

The output of the forecasted results of Fig. 4.4 has been used as an input to PSO to see the

effect on the economics part [120]. The results are shown in Fig. 6.4. It shows the total

reactive power losses for 30 iterations.

Fig. 6.4 Total Reactive Power Losses

Since PSO and GA are two of the most popular intelligent techniques that have been given a

lot of attention in recent years, the proposed PSO method has been compared to GA method

[125]. Both methods have been tested on the IEEE-14 Bus System. Table 6.1 shows the total

reactive power losses in PSO and GA. It shows that with the combination of ARMA, WT

and PSO the losses are quite low and it clearly shows that PSO proves useful towards the

economic dispatch problem in comparison to the GA.

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TABLE 6.1 REACTIVE LOSSES

Methodologies Reactive Power Losses with respect to Slack Bus

(MVAr)

PSO 0.86

GA 2.25

6.7 Summary

The motivation of this research is to put into application wind forecasting studied in the

previous chapters. To do so, the ED problem has been identified as a crucial problem for

which forecasting can be used as a solution to make better dispatch solutions. For this, PSO

has been used in GT to come up with a result, which shows the relationship between various

buses of the IEEE- 14 bus system. The total reactive losses have been calculated and

compared with GA and the results show that PSO is better than GA. Hence, ORPD acts as a

solution for the ED problem that is proved in GT.

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Chapter 7

Summary and Future Research

Recommendations

7.1 Summary

The demand for using various renewable resources arises from changing environment. This

research uses wind as a renewable resource for better integration of wind power into the

electric grid. Intermittency of wind is the greatest challenge and managing it becomes

crucial. The effective solution is to predict the future values of wind power production,

which is totally dependent on wind speed. Changing the prediction model to decrease the

forecast error is one way and has been chosen in this study. There is a need for a hybrid

method that can accurately predict wind speed from a few minutes to several hours.

The types of error have been studied and then a critical literature review and an up-to-date

bibliography on wind forecasting technologies have been done. Review of wind speed

forecasting techniques based on time scale, design issues and types of techniques have been

done in detail. An approach to wind wave forecasting is presented.

It is based on two techniques namely Artificial Neural Networks which are simpler to

construct and have shorter development time and the Ensemble Kalman Filter which is an

excellent technique for data assimilation as it takes maximum advantage of observations. A

fast surrogate is built to train the data for short-term prediction. Therefore, the approach of

EnKF that requires the evolution of the dynamics of the system for large number of

ensembles becomes feasible. Then, their hybrid is constructed and hence proved on

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MATLAB. Combination of ANN and EnKF acts as an output correction scheme. Wind data

is taken on an hourly basis for six hours. The data driven prediction is updated as soon as

they are available using new observations. The analyzed or corrected states can be used for

the next forecast if required. The experiments show that the error can be reduced and a very

good accuracy can be obtained if we use this hybrid for the prediction of the wind speed as

the error is significantly reduced. This corrects the underlying model.

The need arises to come up with a hybrid that can accurately predict wind speed from a few

minutes to several hours. Therefore the ARMA model has been chosen which has proved to

be very accurate for long term forecasting in load forecasting and price forecasting. Simple

input data can be fed into the ARMA but if preprocessed data are fed into it then much better

results can be obtained. The Wavelet Transform has been chosen for this since it

decomposes the data and then reconstructs it. The proposed hybrid is tested with synthetic

data for one hour ahead prediction on MATLAB. Different results are plotted for the WT

and the ARMA model. The WT-ARMA hybrid graph is plotted with the original wind speed

graph to show the forecast error. Comparisons have been done with the ANN-EnKF hybrid

developed in this research. Results show that the WT-ARMA hybrid has comparatively less

errors and less computational time than the ANN-EnKF hybrid.

Also, order estimation of ARMA is of utmost importance. Therefore, ANN has been chosen

and has been combined with AIC to estimate the ARMA true order. The proposed ANN-

ARMA-AIC hybrid has been tested on real data from NIWA on MATLAB. An exhaustive

search has been carried out for different combinations of 1 ≤ p ≤ 6 and 1 ≤ q ≤ 6. The results

have been compared with the Genetic Algorithm that shows no significant difference

between the parameter estimation of ANN and GA thus validating the results. We showed

that the time required to compute estimates of the parameters is less than GA. The results

also showed that if a proper sequence is followed then ANN can be used to estimate the

parameters and in turn AIC can be used to identify the model order correctly. This research

may not only benefit forecasting of wind but also several other applications as well such as

load forecasting or flood forecasting for instance.

The ED problem has been a serious issue in the past and still needs a lot of attention whose

main concern lies in the voltage instability issue. The economic dispatch in the context of

large scale wind generation needs to be reconsidered especially in the context of ensuring

84

overall voltage stability to system operation. In times of excessive or limited generation

periods the dispatch should not be affected. Here the forecast of wind energy becomes very

imperative and might result in the most cost-efficient dispatch option but could be insecure

from voltage security viewpoint. In this research, an ORPD based on PSO has been proposed

to overcome the aforementioned problem. The output of accurately forecasted wind energy

has been used as an input for PSO. The results have been tested on the IEEE 14 bus system

in GT to see the relationship between the forecasted wind speed and its effect on the

economics. The transmission losses are quite low thus proving the amalgamation of these

techniques to be good.

7.2 Future Research Recommendations

The future scope lies in combining ARMA, WT, EnKF and GT with some other techniques,

which are able to remedy the long term forecasting problem. Upcoming new methods like

Support Vector Machine (SVM) that has been tested for few hours’ forecasts can be

combined with these methods to predict day ahead forecast to reduce the forecasting error

even further [110]. This research can also be extended to predict several hours ahead wind

speed and can be improved for medium or long term forecast. This work may not only

benefit forecasting of wind but also several other applications as well such as load

forecasting or flood forecasting. If wind is forecasted properly then wind power production

due to wind energy will increase. Optimization errors can also be decreased when this

research is taken onto another level.

85

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